After encountering Inazuma and breaking no subroutines, I then encounter a piece of Ice with Trace subroutines (eg Shinobi).

While I can't break the subroutines because of Inazuma, does Inazuma stop me from bidding on the traces?

Thanks

No, Inazuma will prevent you from breaking the subroutines on the tracer ICE but will not prevent you from paying into the traces (they will resolve normally).

Breaking a trace subroutine and beating the trace are separate events, and sometimes it is just easier to pay the trace than it is to go through all the steps necessary to break the subroutine to prevent it in the first place (but beware the corps with too much money).