How I handle them in my comic:
I have one mute character in my comic, Bast, but most of the time she's in the form of a cat, not a human which is actually a plot point to how she deals with it. She's mute in both forms of course, but she tends to prefer her feline form as no one expects cats to talk to them. While when you're human on the other hand it's kind of expected that you communicate, and if you have trouble doing so, explain why. So to avoid that, cat form it is.
Being unable to talk was not something that was always a reality to her, she was able to do so before but has since lost the ability, so she has trouble communicating a lot of the time, and in general gets easily frustrated with trying to, though she can express her general feelings or basic things she still runs into trouble with how to communicate, she either not knowing how to properly communicate in a way that can be understood. It's often like a second language, sometimes you're just not sure how to express or even the motion that will correctly translate or express what it is you want to say.
Osiris, the character closest to her. Is often speaking directly to her without expecting a response regardless of her form, and is the best at reading her actions and manners to guess or assume her mood or what it is she is trying to communicate. He speaks for her when necessary, or at least tries to, as he can't always figure out everything she's trying to express and will get things wrong, then of course if she refuses to try and communicate there is very little he can do about that. (which she often does, sometimes it just gets frustrating trying to get someone to understand you to the point you don't want to bother)
My opinion on vocaliser, thought bubbles or telepathy:
While I don't have issues with it, as I've seen it handled pretty well sometimes, I do however think it's a missed opportunity to work with and have a character that can run into these troubles with communication, and just how much it can add to the story and character.
If you're afraid that a character unable to provide dialog would be hard for people to relate to, understand, like or even enjoy, like @CyndiFoster said this character can also be a very good character that a lot of people are going to love if done right. Currently, I have a character popularity poll for my comic going, and Bast, the one who has never said a single word, and only communicated through actions or expressions, is at the moment taking second place.