Roly Nelson wrote:Wood is good, just ask Howard Hughes, who built the largest four-engined seaplane ever flown, entirely out of wood. And that was before Tightbond III had been invented.

Roly
The
Spruce Goose (which is not made of spruce) is impressive, but the material was not chosen because it was better, only because it was available. You can build just about anything out of just about anything (concrete boats, steel airplanes, paper furniture, etc), but there has to be a reason.
If you have a reason for making an trailer structure
entirely out of wood (personal challenge, like working with wood, etc) that's great, but I don't think that the reason is going to be weight savings. Similarly, you can steel-frame a teardrop trailer body, and that's right for some people, but wood is the conventional choice because it works well for the purpose for most people.
Various people have built motorcycles of wood, even with wood suspension components, so a trailer could certainly be done with no structures inboard of the axle spindles and ball coupler in metal. I'm thinking wood suspension arms and rubber springs. Not a project for me, but if someone wants to go for it, I would be interested in seeing the result.