A TRACON Controller reported an aircraft missed a turn to intercept an airway and flew below the Minimum Vectoring Altitude.

Narrative

Aircraft X had departed a satellite airport as was being vectored to join VZZZ. Currently the ZZZ VORTAC is out of service. I was relieving a Controller on position and during the pre-brief I heard her give Aircraft X proper instructions to join the airway. The instructions were vector based rather than to a RNAV fix. The pilot of Aircraft X was simulating the airway with his GPS system; as the needed VORTAC was out of service. A separate safety event occurred in the airspace that consumed the attention of both myself and the Controller I was relieving. That loss of separation and the recovery that followed meant that I nor the other Controller noticed that Aircraft X had missed the turn onto VZZZ. Shortly after assuming the position; I noticed the issue and started to correct it. I told the pilot to turn immediately to avoid a higher MVA. Their turn either due to aircraft capabilities or pilot actions was not able to avoid the MVA area. I did not immediately issue a Low Altitude Alert and this may have lead the pilot to not realize the urgency of the situation. There was a MVA violation resulting from this. In my opinion; this was a pilot deviation that resulted from the Controller unintentionally setting the pilot up to fail. The vector to join the airway while using a simulated airway from an inactive navaid seems quite complex to me. That the pilot failed to do that is not surprising in hindsight. Sending Aircraft X to a fix would have been a better option. I will try to do so in the future. When it comes to my corrective action; I should have immediately issued a Low Altitude Alert. This may have improved the pilot's rate of turn and climb to avoid the higher terrain.