Can you release your zanpakuto if you are mute?
Yes.

Ichigo does it here when he fights Grimmjow





And again when he fights Ulquiorra for the first time in Las Noches





So yes, you can conclusively silently activate your Bankai.

The only other disability I can remember reading of, is Tosen's Blindness. Not only can he wield all of the blade's powers, but it actually adapts to him and his Bankai blinds (among other things) his opponents. No canon source can answer this question about a mute shinigami, none has ever been shown.

Names are important, yes, but it is more about knowing than actually yelling them. Otherwise, top-tier Shinigami fight strategies would be to silence your opponents.