There is a point where a tricky workaround isn’t worth it. 3d vr would require rendering once per eye I’d guess and after that is more required to get it to a headset? No idea. Web browsers have limited capabilities but you can research what’s possible with js since construct doesn’t tout vr as a feature.
Construct just sounds like it would get in the way if all you want to do is relay input and receive output from another program that interacts with the vr headset. But if this is really something you want to try I’d say go for it. You’ll find the feasibility of it as you go.