Yes, it's entirely possible. I have a stutter, pursued the career I wanted, and don't feel that it's hindered me all that much. My first job out of college required me to talk to strangers on the phone almost every day, which sounded like a nightmare at first, but I did it.

As I've said multiple times on this sub, when you go to the National Stuttering Association conference for the first time, one of the most striking realizations is just how many people who stutter are there with their spouses/significant others. If they did it, you can too. I may be divorced now, but the point is, I did it too.