SAT TRACON Controller reported an aircraft departing from 5C1 airport was not in contact with Approach and flew through the final approach airspace of SAT while an air carrier was on downwind to base turn into SAT airport.

Narrative

Aircraft X at 4;000 on a downwind to base turn into SAT. Aircraft Y VFR departed VFR off 5C1 Airport climbing without calling Approach as they fly through the final airspace into SAT.I have reported; and so have many other controllers here at SAT; about the unsafe situation that surrounds the 5C1 Airport and aircraft flying into SAT. For years a Civil Aviation Regulation (CAR) has been open; for years the controllers have been doing briefings to pilots at the satellite airports to inform or educate the pilots as to the hazards around several airports in the vicinity of SAT and nothing has happened. For years the controllers have been dealing with eventuality of an accident happening that will result in many people dying; but the FAA knows this and still nothing has changed. So I guess we'll wait until a series of unfortunate events happen and two aircraft come together. Then; maybe something will change but until then business as usual. I was not the controller in position but watched from another position.