Didn't exist at the time, but the AMRAAM doesn't require the launching aircraft to be using (or presumably even have) a radar. The launching aircraft just gives the missile the target's state vector before launch, which it could have gotten from another aircraft. The AMRAAM then flies to intercept the target, receives course updates if they're available, and then turns on its own radar when it gets close.

The state vector could even potentially come from an AWACS, although the angular resolution wouldn't be great. If the launching aircraft had an IRST system though, that is easily solved.

Incidentally, this basic capability is the present/near future of air combat. Modern but non-stealth aircraft hang back with their radars on. 50 miles or so in front of them are the AMRAAM equipped stealth aircraft getting datalinked targeting from the "quarterback" aircraft behind them. Tracks can be refined by having multiple radars observe the same target from different angles, and further augmented by the stealth aircraft's IRST system if it has one. The (stealth) launching aircraft can then immediately turn tail and run while the quarterback aircraft continues to provide midcourse updates to the missile.