Hey, mine did this exact same thing for a while.....at first they worked fine...then when I hit the button I'd hear the relay click, then about 3-4 seconds later it would click again and that would be it until I restarted the car. At this point, the only time my seats would heat was when it was absolutely frigid outside....below 20F, (and that can be rare here in middle TN) which let me know it was a connection issue. I'd use remote start on those really cold days and on first start up they would work...I guess the cold made them contract enough to complete the connection. Then they got to where they would not work at all. After reading and seeing pics on here I unplugged mine and found the melted connector in the exact same places. I ordered the connector for like $5 total on ebay and just now tonight I fixed them. It took me over an hour clipping, snipping, and being careful not to hurt any of the contacts inside, I was able to just pop the melted plastic off the metal, cleaned it with lacquer thinner to remove residue, and one by one placed the wires back into the new connector. They actually work now! Wonder how long it will be before the new connector melts like the old one. I didn't even have to remove the seat!!! There really should be a safety recall on these cars with this system. I filled out a complaint with NHTSA on their website. Make enough noise and they will investigate it.
The new part # is 15326110 It's just the plastic connector, no wires...unless you get lucky and find one at your local dealership in the back room...that old # is 15306171
If you search for the old # it will pull up the new one and say it's the replacement part #. Good luck in your endeavor...it's tedious but not impossible...and think of the money you will save!!