Yes, it's actually legally possible to built your own, but practically, the inspection service makes the administrative part an unreachable quest.
Not so long ago, the best way was ask them the precize requirements for road use (materials, sizes, weights, lights, brakes, identifying plates ...), you comply that and after the control's visite, you got the registration paper and good to go (insurance, cop's control ...).
Very doable.
A common way to avoid the control's trouble and still get the legal paper was to use a chassis from an old registred travel trailer. Get the proper one for the capacity you are looking for, trash all the superstructure (usually mostly rotten), and build you own flatbed or dump on it, reasonably well. The cops were fine with that.
Now you have to brings (from my immediate recall but there may be more):
- the homologation's reports for the main components like the hook system, the drawbar, the axle. As if the different producers would want to give you that just because you're a nice guy.
- a precise calculation sheet for all the parts.
- (and the best) the result of a crash test of the said trailer !!!
That reached a point where a guy building custom trailers for a living gave up the activity (seen on a mechanical forum).