How can I create an accent wall using reclaimed wood?

It looks like they used pallet wood. The one tip I can give you is to paint the wall a darker color first. That way, a white wall won't peek through any gaps...pallet wood isn't always straight and even!

Sure you can! You can use self stick wood-look tiles/planks. You can use wood flooring, planks cut from plywood, even just marker lines on the wall. The easiest way is self stick vinyl tiles you can cut with a utility knife. Least expensive is likely thin plywood - have the store cut it in strips for you.

Yes, even re-purposing old fencing would work. I would paint first though, then apply the wood for the accent wall. Some use old pallets (free) or even left over flooring from so many re-flooring these days and removing carpet that is often a good source.

https://www.hometalk.com/search/posts?filter=wood%20accent%20wall

I used old fencing. Cleaned then stained ebony. We did cut the “dog ears” off of them first. Everyone who sees it, loves it. Now I’m doing the same thing on the ceiling of my foyer. A very small foyer

Depending on how straight/how bowed your wall is that you want to cover, trying to get pieces of lumber that are not level, or uniform size or thickness, to stay adhered to the wall, may not be so easy.

pallet wood , pine 2x4 ? Any wood you can use ..wide shims