I want to buy a server rack that will hold at least 3 servers (the servers are yet to be purchased as well).

We would like something that can be locked so it's not easy to walk off with one of the servers in the rack.

I've never purchased a server rack before and I have no idea who the reputable vendors are. Is it best to buy the rack and the servers themselves from the same company? If not, how do I make sure I have everything I need (e.g. screws, power strips, etc.)?

Racks and servers tend to be made in a form or standard size or sizes actually. What I think you need is a cabinet not a rack. A rack is the most simple of ways to hold a server. You can if you wish get the two at the same time but I'd guess most people would search for best of each and combine.

The rack is sold either bare or with hardware and hopefully the write up on the sales page would help determine.

They all fit the 19" EIA standard, just some the mounting screws are pre-threaded some have round holes that the nuts clip into and some have square holes they clip into.
Sometimes you can buy an instrument/amp rack/case from a band for cheap, those make nice home rack rather than the refrigerator sized 42U. I had one with handles on it and bolted 4 small wheels on it and could just roll it under a desk.
You can also buy the rack rails and bolt them to a box you make yourself from plywood or MDF or whatever. For servers you buy two pair of rails, front and back, network racks are typically one pair in the center or a small C channel.