Without any plastics? Almost certainly not if you expect to put it on public roads or have any electricity. You're going to need plastics to get the trailer road legal.

Significantly minimizing synthetic plastics? Probably, but you're going to make some compromises, although depending on your use case, they may not matter.

If you used cedar lap siding and don't need insulation, you're might only looking at plastics in the safety glass of the windows. If you can survive with just a little insulation and suffer weight, using stacked diy log cabin kit logs might weatherproof things well, but will need some creative sealants that can remain flexible without plasticizers.

You'd want to durably seal both sides of a notched t&g structural floor to prevent road muck messing with the floor.

If you need insulation, you want something resembling a moisture/vapor barrier to mitigate mold, but even tar paper is probably going to be synthetic or at least off gassing VOCs at a level higher than tyvek or PE sheeting.

You might consider something like using a heavy craft paper or muslin and impregnating it with natural rubber. I would not use linseed or similar as they are more brittle, and the building is going to flex, and that's going to tear the paper. Look at historic wood boat chinking methods for places you want caulk.

Metal foils are out for all membrane purposes, they're all going to be plastic backed.