I think you mean 'bladders', not bolsters. The bolsters are the side pads that keep you in position during spirited driving.
Start by removing the seat cushion for access to the air pump. The pump is black with 3 clear air tubes attached. You write that the pump motor clicks, but doesn't run. That tells me that you have power to the switch, and that the diaphragm is ruptured and jamming the pump mechanism.
To repair the diaphragm, it is not necessary to remove the pump, but removal makes the job easier. Unplug the electrical connections, the clear tubing, and remove the pump. Remove 4 small black screws to access the diaphragm. Open the pump and you will find a round rubber device that has been split open. That is the diaphragm. Carefully remove the connecting pieces, noting how they are assembled. With the diaphragm removed, you can make a replacement diaphragm using a piece of old bicycle inner tube. Some have used a rubber glove, but I think you will find that an inner tuber is more durable. Using the old diaphragm as a pattern, trace around the outer edge and cut your new diaphragm. The diaphragm must fit properly into the pump housing, so care is required in cutting around the perimeter. Assemble the pump, and reinstall in your vette. The first pump will take a little time, the second pump will be a snap.