I tried to post a picture of the relay, but I guess it's too big. I have what I assume is a relay mounted on the firewall, next to the vacuum routing sticker. I can't find a reference to it in the manual and I haven't seen it in pictures of other '83 Eagles. It's about the size of a horn relay, with 3 connections labeled "L", "B", and "S"....and no wiring harness laying around to give me a clue. The car does have cruise control, which doesn't work...but no idea if this relay has anything to do with that.
Does your car still have the feedback carb and stock ignition? I might've been for that. My car, an 86 with cruise, had two of them there, one of them was for the computer, so it got deleted. The cruise should not have anything to do with that relay. If there's no wiring for it, I'd get rid of it.
This may help...https://amceaglesden.com/guide/1988_Large_Electrical_Schematics_-_Much_applicable_to_other_years.
I'd have to double check the diagrams but I know in that area the Idle Speed Control Relay lives around there. It lets the computer control the solenoid part of the sol-vac.