Well I don't think any of our modules require frameworks. Are you saying if you try to use one that does require frameworks, then other modules won't work either? I'm guessing you just mean the ones that require frameworks won't work, and if that is the case, then I don't think that is the problem. I'm not sure if some of the modules we have tried have required other stuff, but I have removed pretty much everything that we decided not to use. (both deleted folders via FTP, and removed their database tables)

EDIT: thinking more about Stewdio's advice, I might have left some database tables that I should remove. Is there any list anywhere of what tables should be included with the default XOOPS installation? Is there also a good way to know what tables each module will add?

I'm working on this with cwebb02, and so I tried (part of) your suggestions. We can't install a module to test, because they will not show up on the modules page. We did have some modules installed that sort of worked (XoopsGallery) but I deleted it since it wouldn't work how I wanted it to. So, in phpMyAdmin I deleted a bunch of tables left over from XoopsGallery, and WordPress, since we got rid of these modules, but it still won't let us add new ones. The modules we have installed and working right now are:
NewBB 2.0
MyAlbum-P
eCal
and Polls
These are also the only modules that show up in the 'xoops_modules' table.
We have tried a couple of different modules, so it doesn't seem to be a problem with the modules we are trying to install. Any suggestions would be great!