EMB ERJ 145 ER/LR
Air Traffic Control Fully Certified
Air Traffic Control Time Certified In Pos 1 (yrs) 12
Situational Awareness
ATC Issue All Types; Inflight Event / Encounter CFTT / CFIT
Air Traffic Control Issued New Clearance
Procedure
Aircraft X [was] cleared for the ILS Runway 12R SJC. Aircraft Y was on an instrument approach RNP-Z Runway 12R to SJC. Aircraft Z was on a go around from ILS Runway 12R. I cleared Aircraft X left turn to ARTAQ (IAF) and cleared them for the ILS Runway 12R.  When I noticed their turn to ARTAQ I switched Aircraft X frequency to SJC tower. I was slowing Aircraft Y down to follow Aircraft X.  I eventually slowed Aircraft Y to slowest practical speed when I noticed Aircraft X going through the ILS final for Runway 12L towards Aircraft Y on the RNP -Z. I had separation because Aircraft Y was only descending to 3;000. I was watching to see if Aircraft X would correct. Aircraft X started to parallel the final so I thought they had it. Then I noticed things get worse with Aircraft X. I cancelled Aircraft Y approach and issued an altitude and vector for a straight in approach to stay away from Aircraft X. I coordinated with SJC tower for instruction for Aircraft X.During this time Aircraft Z went around Runway 12L who was also on the ILS. I was vectoring Aircraft Z back to the left downwind. During that time Aircraft Y went into a 4;000 foot MVA at 3;000. When I noticed I turned Aircraft Z to a lower MVA. When I realized the turn was too late I climbed Aircraft Y to 4;000.[I asked] Aircraft Z reason for go around they said it was an FMS problem. I vectored Aircraft Z back for the ILS. When Aircraft X was on final they said they were not picking up the ILS. I asked if they could switch to the RNP Runway 12L. They tried and it was too late so I cancelled their approach and re-sequenced them for the RNP Runway 12L approach. Aircraft X said their reason was that the ILS was not working. I do not know the situation with the ILS. Next time in low ceiling situation I will wait for my first aircraft to be established straight in on the ILS before transferring communications. I could have easily made Aircraft X go around a cleaner operation.
NorCal TRACON Controller reported vectoring an aircraft below the Minimum Vectoring Altitude.