The question of whether it can be full time comes up a lot and there is no simple answer. How big is the city or town where you live? Does it have a lot of franchises (most shopping is done of chains, not single location places)? How far are you willing to travel to do shops? How much competition is there in your city/town?
I would suggest that you start by reading through the 'sticky' threads at the top of the New Mystery Shopper area for some tips on getting started right with your new business and then start registering with companies that offer shops (MSPs). There is a link at the bottom of each forum page to the forum's Official List of Mystery Shopping Companies.