How do I encourage my father who is unsteady on his feet to sit while shaving?

Does he use an electric shaver or a razor blade. That will make the difference, in my opinion. He will need water to use a razor blade and getting that to the table is an extra step and possible a deal-breaker. With an electric shaver, it shouldn't make a difference, but like freqflyer said, old habits are hard to break. Make sure the chair is comfortable and sturdy. A rolling chair might make him feel insecure, like it will roll out from under him while he's trying to sit or stand. Observe him, if you can, and let him tell you what his resistance is.

Old habits are hard to break.   Either your Dad will find the table/mirror useful or he will never use it.   Just mention it once to Dad and lets see what transpires.

I use to buy gadgets for my Dad to use, and later still find them in carton they came in.