How to record the voices of three people in a room?
Save the money for the on-camera mic, it is by far the worst option. Other options include:

lapel or lavalier mics. Wireless lav mics are as you say expensive, but wired lav mics might be the cheapest option. You'll need something that can take 3 inputs, either a multitrack recorder, or a simple mixer to mix it to stereo (I'd mix the two subjects into one channel and the interviewer into the other).

If you use a shotgun you can get excellent sound quality, but you'll need a boom and a boom swinger to make sure that the person talking is on-mic. For an unscripted interview with three people this is quite a tall order. If you don't have someone who can do the job of boom swinger, or if your subjects interrupt or talk over each other this is not recommended.

a surface mic or an omnidirectional mic in the center of the speakers. This won't be as good as three lapel mics, because it will be further away from all of them, and will pick up more room ambience and bg noise. This is probably be the cheapest option, but the second worst (the worst being an on-camera mic).