The Chiesa (Church) di Orsanmichele is still in operation - it isn't just a museum! If you would like to attend a service, the official website of the church is http://www.orsanmichele.org/
It appears from their site that services are only on the weekend:
Orsanmichele Celebrazioni Eucaristiche
Sabato 18.30 S.Messa prefestiva
Domenica 11.00 (maggio-ottobre) 18.30 S.Messa vespertina
There is a phone number and email address also for up to date information:
Phone: 39 055-210305
Email: san.carlo@alice.it
