Can I build a 4A circuit on cardboard?
Yes it would work and yes you might have to consider the possibility of the cardboard lighting up under high loads. If you do it right you might be fine. Maybe consider building a boardless circuit. Look up "flywire circuits" and you will see examples. If you use heavy solid wire for you ground you could even incorporate mounting loops in the wire. It will take a little more skill but your results can be beautiful,  last for years and be fireproof.