A Tower Controller reported they issued an aircraft on ten mile final a left 360 delay vector which place the aircraft below the Minimum Vectoring Altitude.

Narrative

We were operating on Runway XXL and Runway XXR which is the standard configuration. The traffic volume was high that we split the local control position; one controller working traffic to the south parallel; Runway XXL; and one controller working pattern traffic for Runway XXR. I was working the local control position sequencing aircraft to Runway XXL. Taxiway was on the RNAV approach for Runway XXL. The pilot checked in on a 10-mile final stating that they had yet to complete a certain checklist and requested a vector to do so. I coordinated with the other local controller and proceeded to issue a right 360 for their vector. In doing so; I removed the aircraft from the RNAV approach without having issued a visual approach and ensuring the aircraft had the airport in sight. As the aircraft was completing the 360 I went back to issue the visual approach and ensured the aircraft had the airport in sight which the pilot responded back in the affirmative. As the aircraft was in the 360; the Supervisor called the overlying TRACON to advise them that the aircraft had been cleared for the visual and was issued a vector. The aircraft landed without issue.At the time of the incident; I was not aware that I had vectored an aircraft below the MVA in this incident. It was not until a few days later that my direct supervisor informed me that this was a potentially significant event. After speaking to my Supervisor; we discussed different ways to handle this situation if it were to occur again. The first option is to ask the pilot either to begin the published missed approach or to cancel IFR and fly VFR given that the pilot reports they can keep the airport in sight. If the pilot insisted on a vector; then begin to clear the pilot for the Visual Approach by ensuring they have the airport in sight and then the Visual Approach clearance. While all this is occurring advise the IFR authority of the pilot's intention.