A TRACON Controller reported a departing air ambulance failed to turn as depicted on the SID and flew below the minimum vectoring altitude.

Narrative

Aircraft X departed ZZZ Runway XX via the ZZZZZ departure. I noticed the MSAW going off immediately; but they were on the upwind and the MSAW went off at random times in and around ZZZ. I read the remarks on the strip and read Aircraft X thinking that was their call sign. I should have known better and feel like an idiot for calling them Aircraft X. I issued a low altitude alert and an expedited climb to 19000 ft. once I knew they were in an unsafe situation. I thought with radar delay that maybe it wasn't showing their turn yet and ZZZ per our Letter of Agreement (LOA) is supposed to call us if they are not in their turn by a mile from the departure end. I received no call and told them to turn toward ZZZ1 where there is lower terrain and like they were supposed to. Once they met my MVA I put them on course and then was relieved; the next Controller issued a warning.It's crap that after working 6 day work weeks and only 2 Controllers on break that this is going to be written up like it's all my fault when I am by far the most vocal person in ZZZ trying to get supervisors and OM's to somehow make these procedures flown right. Everyone keeps thinking its only the ZZZZZ1 departure but it's now the ZZZZZ; ZZZZZ2; ILS to Runway YYL; ZZZZZ3; and ZZZZZ4 that I've had pilots try to fly into mountains and our management has done nothing to get it changed. I'm a member of the Organization and bring up ZZZ every month and nothing goes further than that. Our upper management is worried about whether relieving Controllers verbally state they have the approach or if we have more than two Controllers on break instead of worrying about safety. ZZZ is a toxic workplace. What should be happening is our facility needs to talk to the airlines and companies that update the autopilot systems in the system. There is commercial airlines saying they can't load departures and arrivals properly; but we are just letting it happen.