Caravan Undifferentiated
Air Traffic Control Developmental
ATC Issue All Types; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter CFTT / CFIT
Air Traffic Control Issued Advisory / Alert; Air Traffic Control Issued New Clearance; Flight Crew Requested ATC Assistance / Clarification
Human Factors; Airspace Structure; Procedure
Aircraft X was out of approaches airspace. I cleared him for the VRA approach. On the airway once he reaches the IAF; he's suppose to start turning southeast bound on the approach course. I noticed he passed the IAF and I asked Aircraft X if he was on the approach course. He replied we are correcting. Then; he came back and asked if I could clear him to another fix which is the final approach course and I said unable. I then asked if he needed vectors and he said yes. He was in a 4;000 feet MIA (Minimum IFR Altitude) and at 3;200 feet. I replied 'Cancel approach clearance remain on the final approach course; climb and maintain 4;000.'  Then I also added reaching 4;000 [feet] fly heading 210 vector for the ILS. He started climbing and once reaching 4;000 feet he started his turn for about 2 seconds before saying he had the airport in sight. I then gave him the Visual Approach.I then started asking questions about what I can do different in this situation and not put myself at risk of an event. I believe the best course of action would to have been ask if he could maintain his own terrain and obstruction avoidance and if he couldn't then treat it like an emergency. I originally was trying not to vector below my MIA but I will do as described above if the situation happens again.
Trainee Controller reported they observed an aircraft cleared for an approach fly off course and below the Minimum IFR Altitude.