I need to reinstall my instance of Ableton, but I don't want to lose my plugins and their organization.

Is there a simple way to do that? If I replace my old instance of Ableton with my new instance will Ableton write new empty folders in the place of all the old ones?

Thanks for the advice.

That's what I was thinking, but aren't there like a bunch of other random folders in the audio/library section of our computers that carry some specific DLL files for plugins? Maybe I'm overthinking it and there's really just a few.

The question is whether you mean just plugins, or not. If you're worried about VST plugins, then do what outofplastic said. If you also mean rack presets, Ableton instruments presets and stuff like this, you'll probably want to backup Ableton's library folder too. ´

...aslo, dunno why you need to re-install, but there's this little neat trick that cures a lot of trouble with Ableton.
Go to C://Users/(your account's name)/AppData/Roaming/Ableton/(Live version you use at the moment)/Preferences
In that folder, find file "Preferences.cfg" and delete it (or for backup reasons just move it out of that folder). It stores your Ableton's settings. If it ain't there, Ableton notices it and creates fresh new one with totally initialized settings. Try that, it might cure whatever you're solving.