TRACON Controller stated a VFR aircraft departed 5C1 airport and climbed into confliction with an air carrier on an ILS Approach to SAT.

Narrative

Aircraft X was turning base for ILS approach into SAT and descending for 4000 ft to 3000 ft. We observed a target near 5C1 that was at 2500. We did not know what the unknown traffic was doing or what they were climbing to. Previous aircraft off of 5C1 turned NW bound this aircraft went NE and started to climb. Aircraft X was issued traffic and instructed to maintain 3500. Trainee answered a line from ZHU [Center] when the Captain started going off. Trainee issued some form of a TA and a turn to avoid traffic.Extend Class Charlie airspace and have aircraft departing 5C1 that climb above 2500 ft. contact SAT approach. If you ask every Controller at this facility where is the highest chance for an aircraft accident at this facility will be most if not all will point to 5C1 aircraft being allowed to fly on final without talking to approach. This is an extremely dangerous situation to the NAS and we locally have been shouting for over 10 years for change. Nothing will happen until there's an aircraft accident.