I work from home a lot. I am a computer programmer, and with all the communication technologies out there, it doesn't matter so much where I am. I'm working with teams in New York, Seattle, and various parts of SF right now.
In my profession, working from home tends to be a perk you get if you can prove yourself beforehand.
I THINK that it would be illegal for a company not to hire you because of a speech impediment, assuming that it wouldn't harm your job performance. As you said, cold calling may not be in your future.
What other jobs have you held in the past?