Is there a scanner or printer that would enable me to quickly scan tons of old family photos that would have some sort of auto feed mechanism for 20-30 photos at a time? I see some really pricey scanner options but would prefer something that might be under $200 instead of 500+.

Also concerned because it looks like a lot of printers may have a scanner with a document feeder but it would have to bend the photos around a roller to spit them back out. Not sure if that is an issue or not.

The feeder requirement is what will drive up the cost for you. Your budget would get you some good photo scanners, but not with a feeder.

You might try Google PhotoScan. Using just my phone, I have converted many old family photos with this. Easy to use.

If it's a large number above a few hundred, I think its worth it to have someone else do it. They'll be able to give you high res pics and do it much faster