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That is a tough one. A stuck closed relay can cause compressor failure, but I'm not sure if you'll be able to convince the repair shop to take responsibility for it. Most of the time the compressor will start venting off refrigerant when the relay gets stuck closed and won't necessarily damage the compressor right away. If you drove for a long distance with the relay stuck it might have been enough to purge all or most of the refrigerant off, then without refrigerant to act as a lubricant it could definitely start to seize up.
I would probably take all of the documentation in to the shop that first changed the relay and see what they will do. It's possible that they have a good parts vendor that will take care of the compressor cost under their warranty.
I do agree with Toyota- I think the relay was probably the cause of the compressor failure.
If you do end up in small claims court, my opinion (I used to work as an expert witness for small automotive claims cases) is to not push too hard on the aftermarket vs OEM parts argument. The Magnusson-Moss warranty act states that if a specific brand of part is required to make a vehicle function properly then the manufacturer of that vehicle has to supply the part at no cost. Some judges will throw it back on you saying if the wrong brand of part is the cause of the failure then your argument is with Toyota- good luck getting anywhere with them on that.
I would probably just argue that the part they installed failed prematurely- most likely it was defective- and leave the OEM vs aftermarket out of it. Unless of course the shop used a non OEM quality relay, which you don't really see much of anymore. Most parts manufacturers even aftermarket only make OEM certified parts nowadays.
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