Not really, but it can be involved for some tasks. 'Muscle Memory' is a bit of a complex thing. It's not so much the firing of the neurons, as much as it is the timing of that firing. Your reaction time is at the ~5ms level. Much longer than the muscles need to move in concert to, say, hit a 3-pointer. Controlling all of that can take place all the way from the brain down to the ganglia of the spinal chord. Drinking a cup of tea while reading will mostly take place before the brain gets a chance to intervene, for example. While riding a bike will involve more of the brain. I want to stress, it's a complex and not well studied area of active research.
It is interesting though. Do you know if there is a name for this particular area of neuroscience?
There is not, it would just be general neuroscience. I'm unaware of specific labs either. Google would be your best friend in terms of trying to find specific researchers and in reaching out to them.
I’ve many times had the experience of trying to debug someone’s computer problem, and trying to describe how to fix something, I couldn’t think of what to do in words. So I said, “my hands know where the answer is” and once I had the mouse I clicked around and did the task fairly quickly. I wonder if that was the cerebellum solving the problem for me?