If I have corrupt files and want to detect whether the Flash NAND SSD is the issue or whether the installation is just corrupt, would scanning for bad sectors help, or is this just useful on regular hard disk drives?
Generally speaking, SSD's firmware should be taking care of re-locating your data if the current location is about to go bad. However. Nothing is 100% reliable, including flash firmware re-allocation algorithms. I'd say the chance of real bad sector (e.g. you write data, but cannot read it back, or read back not what you've written) on SSD is pretty low, but it won't hurt to scan for bad sectors either.
If some non-trivial amount of bad sectors do turn up on SSD I wouldn't use that SSD at all though, it means something is seriously wrong with either its firmware or the flash NAND itself.
I would first suggest querying the drive's SMART parameters. For example, the drive might have exhausted its supply of spare memory blocks. You'll want to use the most recent version of smartmontools you can easily get your hands on (one easy/free source: an Ubuntu LiveCD.)
Smartmontools can also activate a drive self-test and show you the device's smart error logs, if any.
If SMART shows no issues, then it's probably not "media" issues (ie bad NAND cells.) Corruption might have been caused by a firmware issue; flash drives have often seen many firmware revisions, so check for updated firmware for your drive.
To answer your original question: yes, you can run badblocks (or similar) on an SSD.