I came on here to see if there is since I don't use this site at all anymore but appears there is no option.
I wanted to delete an old account of mine myself. Even if you can't fully delete it I at least wanted it closed or something. However, I can't log in. I requested to have my password reset but didn't even receive the email!
"Success! Provided the supplied email matches a user account in our records, information on how to reset your password has been sent to your email address."
NO IT HASN'T!
Link
Gamedog
https://www.weasyl.com/help/faq#account-deletion
"We are not able to fully delete Weasyl accounts at this time. If you no longer wish to use your account, we recommend that you remove any personal information and other content from your profile before signing out."