If you haven't logged into Neopets for a while...
... you may be asked to provide your birthdate in addition to the password. And if you haven't logged in even longer, then you may be forced to do a password reset for safety reasons. In that case, simply go to https://www.neopets.com/account/passwordreset.phtml and provide your username (one account per ticket!) and click send.
IF the email goes through, simply follow the instructions it provides to continue unlocking your account. But more often than not, the email will not be sent for whatever reason. Or you may have used a throwaway email address to create the account, or you no longer have access to the email in question.
Also, always check the spam folder! If you maintain a block list, then whitelist anything related to Neopets before submitting a ticket or resetting passwords.
If you still didn't get a response after 72 hours or you don't have access to the original email for whatever reason, then you need to submit a ticket instead.
What is the latest account recovery policy?
From the site:
As of September 2023, the account retrieval process will use updated security questions to verify account ownership. The number of security questions required during the retrieval process depends on whether the account is classified as standard or sensitive.
Standard accounts are those held by the average Neopets player. These accounts are unlikely targets for theft and require basic security questions to be answered correctly during the account recovery process.
Sensitive accounts are those with large amounts of Neopoints or NeoCash, unique Neopets, desirable usernames, or other factors that could make the account a target for theft. Sensitive accounts have enhanced security measures in place and require an additional set of security questions answered correctly to regain access.
If users are still unhappy with the outcome after going through this process, they now have the option to request an appeal. A Customer Support Supervisor will carefully reevaluate the details of your case and provide a comprehensive resolution within 14 business days. Be sure to include your original ticket number.
Submitting a ticket:
https://support.neopets.com/hc/en-us is the on-site ticket creation help. In that site, you're presented with a bunch of questions / options providing more information about possible issues with the site, but in order to actually submit a ticket you have to scroll down to the bottom and click on the yellow button here:

(Direct link: here)
You can only view your ticket number after you got a confirmation email saying that your ticket has been opened and is under review, which can take few days. The support team is chronically understaffed so please be patient.
Highway to Help:
You might be able to speed up the process by commenting on their facebook account (you can just create an account, comment once and then log out forever if you don't feel comfortable having a FB account). Provide your ticket number then, but no other information, as those comments are public! Adding more sensitive information like the username and names of your pets to FB may backfire as anyone else can then try to pose as yourself and retrieve your account.
What information do they need from me?
They will ask you about the account in question, and they usually want at least the username, birthdate and associated email addresses. If you used a throwaway email address[^1] to register, and/or used a false birthday (because let's be honest we all got told to never share private information on the internet as kids....) then write those down! Take screenshots, maintain a word document, note them in your diary, tattoo them onto your forehead - anything as long as you have the most important data ready for when you need it.
This is the most basic information they're asking for, but if your account is a "sensitive" account (10+ years old, lots of NP / trophies, unique/rare pets, anything a hacker would deem worthwile) they may ask for more information than just the basics.
Don't be discouraged by the humongous list below: you won't need to know ALL of these, and neither will the support team ask for literally everything. Those are just questions that MAY be important for the NP team to determine whether or not you're really the owner of the account you're trying to unlock.
If you still HAVE access to your account and you haven't saved any important information yet, I'd advise to start doing that ... just in case.
Information about sensitive accounts:
About the account itself:
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When was the account created?
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What's your displayed in-game name and/or gender?
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What's the country you selected?
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Do you have a special message and/or theme in your profile? What language is it written in?
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Who are your neofriends and/or VIPs?
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What guild did you join?
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Have you ever recieved warnings, and if yes, what were they about? Was your account ever frozen (for protection)?
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Did you redeem rare item codes, and if yes, which ones?
Achievements and Trophies:
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How many / which avatars do you have? (Especially hard-to-get ones or those only availiable for a limited time, like from unique site events)
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Do you have rare stamps, keyquest tokens or TCG cards?
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Trophies and medals, like the one from site events or recurring challenges (Altador Cup, Charity Corner etc.), hard-to-achieve game trophies or ones for random events
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Do you have access to the lab ray?
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Which Battledome Challengers have you unlocked?
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Do you have beads on your Lutari Talisman?
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Has your account or one of your pets been featured in the Neopian Times?
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Which stocks have you invested in? Which ones do you currently own?
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How many NP do you have in your bank account? (it does not have to be 100% accurate, but it does make a huge difference whether you own 50 NP or 50 million NP)
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Which auctions and/or Trading Post offers have you won OR put up yourself?
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Have your accounts and/or pets ever won contests? (like for example the lottery, gallery spotlight, or customization spotlight)
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Do you have access to the Magma Pool? (Not just sneaking by the guard, but official access)
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Do you have active and/or used-up Fountain Faerie Quests? If you used one, which pet and what did it turn into?
Pets (and petpets):
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Species, name, gender, and whether you created them yourself or adopted them from other users / the pound.
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If they have been zapped, painted or morphed, then also note the species and color they started with.
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If you moved them between accounts[^2] then jot down from which account the pet originally came from, when they were adopted, or usernames from people who adopted your pounded pets.
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Each pet also has a petpage, so if you happen to have written something special on said page (like game guides, an art gallery or something like that) it might be worth mentioning.
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Have your pets been trained / zapped and ended up with high stats? Did you unlock Fairie Abilities, and if yes, which ones?
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Which Gourmet Food and/or Booktastic Books has your pet consumed?
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If you have played Neoquest 1 and/or 2 then the save data is tied to your active PET, not to the account! Keep track of which pet and how far you have gotten in the minigame.
Unique/special items:
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Battledome equipment currently worn by your pets
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Stuff stowed away in your SDB like rare petpets and petpetpets, paint brushes or NC-item leftovers from events
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Items in your gallery (also the name and size of the gallery and/or shop you have)
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Paint Brush Clothing, expensive/rare wearables, expensive / rare Morphing Potions
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NC and NC items, like Training/Quest Cookies, Time Capsules, wearables (especially when it was something only availiable for a limited time), Goodie Bags, Cupcakes (from pet birthdays or NC mall events) and whether or not you bought NC yourself or "just" won it from the Expellibox / got it as gifts from events
Heads up!
If you don't know what some of these mean, just ignore the question. Not everyone HAS beads on the Lutari Talisman or bothered feeding their pets with Gourmet Food, so don't fret about it too much.
Also, it is unlikely that they want to know ALL of this at once anyway, so please don't just copy-paste 57 pages worth of information into the initial email. Stick to the questions they asked first, and once they ask for additional information, then answer those questions as well and provide some extra info you deem important.
And always, ALWAYS attach the rest of the conversation to the ticket response. Otherwise it might happen that they ask for information that you have already provided in earlier emails, keeping the ticket in repetition limbo.
What you should NOT tell them:
What they will NEVER EVER ask for, is the current(!) password. If you get an email asking for your password, it is almost guaranteed to be a scam attempt. They might ask for passwords you have used in the past as long as they are obsolete / changed now, but they will never ask your current one. Same for a PIN if you have one for your shop or SDB.
Never. Ever. Tell. ANYONE. Your. Password.
Need additional help?
As someone who had to reactivate his accounts multiple times after various hiatuses, I'll be glad to walk you through the process if you should need additional information / help.
Feel free to ask!
Footnotes:
[^1]: Whether you want to have one single email address for all accounts or one unique email address for each separate account, that's entirely up to you. A single email address for all accounts makes retrieval easier as the NP team can see that all (up to) 5 accounts belong to the same person easily. Having different addresses might lead to issues while retrieving the account, but is overall safer for when your accounts may be targeted by hackers. Only "temporary" email addresses are an issue as they might lead to the account becoming un-retrievable, so change those to something that won't get deleted ASAP.
[^2]: You're allowed to have up to 4 side accounts as long as you solely use them as item/pet storage. You are not allowed to earn NP, obtain items from dailies, or play games that may lead to trophies/prizes on side accounts.