Is it possible currently or theoretically to build a supersonic fighter jet from non-metallic composites?

Currently it's not entirely possible, although wings and a lot of the overall structure is already made of composites.

It is theoretically possible to build an entire craft w/o metal, but it is quite unlikely that it would happen.

Some parts are more beneficial than others to build out of composites, and some parts are outright difficult to build out of composites.

And believe it or not, but the military is actually concerned with cost. If metal is the obvious choice from engineering reasons, they wouldn't want to pay extra to get the part made from composites just because.