Is this a accDB file, or a mdb file? That could well matter. However, as noted, I doubt much ULS security is being used, else when you logged in as admin, and it works fine, then that again suggests this is not ULS. As noted, not practical to re-write the software, but if there is only 1-3 pages that use the database, then as noted, you can continue to use Access, but replace the create object ado, with .net ado. If one is familiar with both object models, then the efforts to do this change is minor (depending on how much code)..
As noted, this is simply the environment and deck of cards you been dealt with - so narratives around changing this or that code is not a whole lot of help to you. As suggest, you might want to try creating a local user on the server, and specify that for the app pool, and not a domain user. And since the site runs with you as admin, then again, this does not look to be a bit size miss-match. As noted, using create object is much the same as using a COM object, or so called ActiveX, and out of the box, that option is locked down on a server. So, try a quick test with a local non domain user on the server -- that would be my best idea to try, and it only takes a few minutes at most.
So, create that local user on the server. Logout, and then logon using that user, and see if you code runs on the server. If yes, then a good chance that external uses to the server will also now work.