Is Shimano Claris compatible with disc brakes? All of the Shimano Claris groupsets I have seen come with cantilever brakes as far as I can tell. 

Is it possible to get this with disc brakes, would I have to buy the individual components?

What would be a good, affordable configuration of components (or entire groupset ideally) to get this? 

I'm thinking of STI levers, is this important?
You can use disk brakes with road levers as long as you buy the ones specifically designed to work with road levers. The difference is the amount of cable pull (the distance that the cable moves when you squeeze the lever). This distance is lower for standard road levers than for MTB levers, so the design of the brake has to take this into account. I know that the Avid BB7 brakes are available in road and mountain versions, I'm not familiar with the Shimano line.