I tried using a relay to turn it all on. but the relay wasn't getting
enough power. A friend mentioned a solid state relay, maybe that will
work.

"Typical" solid state relays generally have input impedances of between 1k
and 10k ohms. The relay I'm suggesting from radio shack has an input
impedance of 400 ohms. That should be fine. I'm thinking you may have
tried one of the larger radio shack relays or bosch relays where the input
impedance is under 100 ohms.

The buttons in the front of the car are just that: Buttons. The
accual reciever is located in the trunk. I have a 1997 Ford Taurus LX.
The unit in the back has a small bar code with no numbers on it and
"JBL" written across it.

Ah, I see. I've done installations in Probes that had a powered JBL system
in it in which case I've just replaced everything. That may not be the
route you want to go in, huh?

I believe the smoke I saw came from the left
rear speaker. When I tested it with a meter it went to zero Ohms, so
that speaker is shot. I don't know why there would be smoke when a
speaker blows out, I've never experienced it first hand before.

Well, consider that a speaker often blows by melting of the
solvents/laquers/adhesives and sometimes the vc itself. That'll make some
smoke, but I'm still surprised you saw some.