I really don't remember from the book, but in the movie, Snape viciously aims his wand at Harry and yells, "Legilimens!". After which Harry screams at the intrusion into his mind.

I don't get the impression that either one requires a wand. Snape is adept at both and seems to do it wandlessly. It would be hard to occlude your mind secretly if you had to use a wand. It would be obvious to everyone around. I never saw Voldemort using a wand when he tried to pry information from minds with Legilimens. My thought is that when he commands Harry to 'clear your mind' he uses his wand to needlessly torment him with images that rip through his mind and cause him as much discomfort as possible.

So, the answer is a qualified, 'no'.