There isn't going to be any inventory BUT there WILL be houses for sale in your budget in Columbia. As someone who just bought a home in that price range, it feels scary because there is 0 inventory, but there ARE houses selling in that range.

Some of the neighborhoods people are mentioning here (near the library, east campus (buying not renting), Westwood area) are going to be way out of your budget. Around Columbia College and Benton Stephens are close to downtown but with some affordable [older] homes. I think of anything near Stadium (or within the loop it forms) is pretty central.

Some people on this sub trash the north side a lot (north of I70), but I think it is neighborhood by neighborhood. Some neighborhoods are super isolated (but not more isolated than the far south neighborhoods and FAR less expensive), some are really nice and close to parks. It isn't really possible to walk to downtown or campus from north of I70 though.

This isn't unique to Columbia, but houses go fast so the best way to see them is to get a realtor and jump immediately on anything you see on Zillow so that you can see some houses. I would look at houses first thing in the morning, immediately text my realtor if I was interested in any, and try to see them by the end of the day.