So I dont have any soruces to back this up, this is just supposition. The answer likely varies chapter to chapter. Certain chapters might have a single house that they procure services from, whereas others maintain ties with multiple houses. Other chapters, such as the Carcarodons, might acquire their Navigators from a variety of services. We know that the Carcarodons operate on the fringe of the Imperium, where resources are difficult to acquire. They are known to indenture Imperial citizens as they see fit, even a stranded Rogue Trader. It's likely part of the Red Tithe that they 'conscript' navigators.
As for what happens to the Houses associated with renegades, I'd imagine that even the Inquisition treads lightly when investigating Navigators. The Navis Nobilite holds significant power in the Imperium, and their role is vital to interstellar travel. I'm not saying that Inquisitors wont investigate them, but they'd likely need more then circumstantial evidence to effectively prosecute them: even Inquisitors require the services of Navigators.