It certainly is possible, but it would be hard to start one without prior reputation. The hardest part will be trying to find companies who trust you enough to allow you to grow through them; taking extra time because you need to borrow test boxes or order parts for instance. But that being said, my company was started by two techs who wanted to do things the right way and we are doing very well now! Of course there are growing pains, but overall if you have the dedication I'm sure you'll find success.

In US, all avionics work is completed by mechanics (or A&P [Airframe and Powerplant] technicians). I'm sure there are specialized shops and nuances that I'm missing, but as far as official titles go, in the US it all falls under aircraft maintenance as far as I know.