Can someone become a medical assistant w/o the class?

I know these career schools crank out MA's like nobody's business, but I am wondering if it would be possible to be hired as an MA just with on the job training. I have a BA degree in another area that I am not using, have a CNA certificate, and have taken medical terminology as well as all the usual nursing pre-reqs. I am looking to advance to higher pay and a greater scope of practice, but am in a deadlock until I get accepted somewhere for nursing school. Would something else like plebotomy be a better option to combine with my CNA certificate for my temporary career? What if I took free computer certification courses at work to learn various microsoft applications? Other thoughts?