Can you release your zanpakuto if you are mute? So, this sounds like a stupid question I know but in the bleachverse 'names' are very important and you can only unleash the power of your zanpakuto if you can call out its name. Therefore, if a shinigami is mute, will they ever be able to achieve shikai/bankai? If so then how? Only one case that I know of is for shinigamis who have already achieved bankai to be able to release their shikais without a command.

Answers based on LN are welcome, but if you are citing from a dubious source or a source whose canon status can be question, please specifically mention it. I am mostly interested in an answer based on a  canon source.
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.