Yes, you can :) https://www.uncarvedblock.com.au/build-climbing-wall-at-home... gives you a bit of a guide, and if you search climbing walls on youtube, plenty of people have been building them.

Mine is framed in 70x35mm pine, and built from formwork plywood (formply), which is 17mm thick. The holds attach with 3/8th inch bolts that go into t-nuts in the back. I still need to give the formply a bit of a sand and put a non-slip coating on it, as the holds can rotate if I've not recently tightened them.