Yes, in order to short you must have a margin account.
However there is another factor. The other part is that the stock must also be on your brokers short list for that day. In order for you to go short, you are always borrowing against someone elses long position. This borrowed stock can be held by another account at the same clearing firm or a clearing firm that they have a reciprocal borrowing agreement with.
There will definitely be occassions when one broker has a stock shortable and another does not. Gerneally speaking, it doesn't really matter too much which broker you use, because most of them use the same few clearing firms or their clearing firms reciprocal relationships are likely similar.