Mormon missionaries or just regular Mormons might be happy to help. It doesn't
count against [our] service time in the same way that door-to-door proselytizing does
(as there is no such requirement, service hours aren't recorded anywhere, and only our full-time missionaries do door-to-door proselyting). We tend to be a helpful bunch, so you could go ahead and ask. However, I would be puzzled if a random guy called me up to ask me to help him move. Although we do often help people move, we are regular people and are not movers. Pizza might help the situation, but if you call the missionaries be prepared to be talked to about religion (worst case scenario it is a bit awkward). If you want help getting into contact with the missionaries to set something up, you could PM me or just Google it.
I also know that college students will do just about anything for food. When I moved I asked my brother to round up a bunch of his college buddies. They were excited about free pizza and ice cream. I'm sure if you get creative you could get something to work. Just be warned though, if they aren't pro movers, they might break your stuff because they won't necessarily be careful with your stuff.