For me, "taxi cab" is redundant. I'd use one or the other. Cab is a bit more informal. Also, I've reworded the sentence to put the emphasis on the profession of being a taxi driver. It is possible to work as a taxi driver in New York without a taxi medallion?

Exactly, why is it not a language question?

I guess you mean: "Is it possible for someone to drive a taxi cab in New York without a taxi medallion?" But that's not a language question.