Hi, I have a vintage bike with Shimano Tiagra ST-4501-9 integrated brake lever/shifters and unbranded brake calipers. I'd like to change the brake calipers but I can't find ST-4501 on the compatibility charts. Will they be compatible with BR-3500 calipers?

I'd post pictures but I haven't had 10 posts yet.

As a general rule, most road brake levers (and STI's) can operate most typical caliper brakes.
Whether a brake will work on bike has not to do with the brake levers, but the frame measurements.

What you need to look for on your new calipers, is the 'reach' of the arms, as in how far the brake pads / wheel rim are from the bolt where the caliper mounts to the frame.

Usually, the bigger the tires the bike has on it, the longer reach brake it needs.

High-end, racing calipers tend to be short-reach; Something like the 3500 Sora calipers you picked are mid-reach, with a fair amount of adjustability.