In an MDB format database without ULS (user-level security), user operations are performed as the Admin Access security account user.
In your situation, it seems you have a user --- let's call him Fred --- who created a table named tblFred. But now you are unable to delete that table.
Even though you have not applied ULS, Access still maintains user and group permissions for database objects. And it sounds like somehow the AdMin user has lost the necessary permissions on tblFred.
In order to delete the table, Admin needs DROP permission for tblFred. And if the table contains any data, Admin also needs DELETE permission.
Go to the Immediate window (Ctrl+g) and GRANT Admin those permissions.
CurrentProject.Connection.Execute "GRANT DROP ON TABLE tblFred TO Admin;"
CurrentProject.Connection.Execute "GRANT DELETE ON TABLE tblFred TO Admin;"
I'm unsure about precedence for those statements, so repeat them in reverse order if the first attempt doesn't allow you to then delete the table. And hopefully that effort restores your ability to delete the table using your normal method.
You may want to inspect permissions on other database objects to look for inconsistencies. (And actually you could use this instead of executing the GRANT statements if you prefer.)
DoCmd.RunCommand acCmdUserAndGroupPermissions
However, even if those suggestions worked, I can't explain why the permissions got broken in the first place. As only a guess, I'll blame the fact that you have multiple users sharing the same front-end db file. Although your data is housed in a separate db file, Access still maintains state for operations in the front-end sessions. So when multiple users have the front-end open, things can get confused, and odd things can happen. That is why the safest practice is to give each user their own copy of the front-end db file.
I just read the question and comments again. I had assumed you're using a split application, but you didn't actually say that. If the users are sharing a single db file which contains the data and the front-end pieces (queries, forms, reports, etc.), the situation is more fragile than I assumed. Bad things are bound to happen.