Generally, it shouldn't be a problem.
Wood moves very little along it's length (almost zero). Make sure it's sealed and painted well, and you shouldn't have any issues at all.
My router is almost 15 years old, and made entirely of wood, and is the same today as when I built it. It's in my garage, which gets down to freezing in the winter, and over 90F in the summer. I do heat it in the winter when I'll be working out there, but I don't keep it heated.
I'm building a new 4x8 machine which also will be almost entirely made of wood. I expect it to be as good or better than a CNCRP machine, with better performance.