I don't have an answer for a part number for you, but the parts counter at a Toyota dealership would. Or, you can call 1stToyotaParts.com (877-810-7067), which is on the west coast, Monday - Friday 7:00 AM - 7:00 PM or Saturday 8:00 AM - 4:30 PM pacific time. They have always been very helpful to me giving me the part numbers and checking on availability of parts through Toyota. They also sell parts at wholesale prices, not retail, so you get a better deal from them than at your local dealership.
When I put your part number in the 1stToyotaParts.com website, it has a number of applications for this relay, one of which is a cooler relay on 1986-1989 models, relay R (but it doesn't say which Toyota model). When I look up your specific year & model, it says 2 relays are used, this particular part number (relay R) and another relay, called relay G. So if I were to "guess", I would think you have the right relay for the right application in your Celica. It is a pricey relay ($52.64 retail, $37.98 thru 1stToyotaParts.com).
If the function schematic is printed on your old relay, why don't you short circuit the 2 connectors in the relay block that would provide power to the AC compressor clutch, and see if you light stays on steady and your AC system works as it should (but don't keep it there too long, because the compressor will not shut off with you short circuiting it). If this works, you know you have a defective relay, or you have a signal wiring problem that is used to activate the relay. But at least you can narrow it down.