N6ICW does. Full time coverage on 147.405, located on Mount Vaca, with a a directional antenna to the northeast. This can be heard about one hour before launch to about a half hour after landing.
The repeaters will also rebroadcast this audio on a part-time basis (take-offs and landings). 147.195+ which is simulcasted from Bald Mt. (Yolo County) and from Mt. Vaca. It has 6 voting receivers in and around the Sacramento area.
The 145.150 repeater is located in South Lake Tahoe.
I think most people want to hear the shutttle launch and not people talking. I'd stay clear of transmitting while they are rebroadcasting the launch audio. I would imagine you could talk..but you can just go to any other repeater and use it. There's plenty around our area that are not rebroadcasting it.
N6ICW might even disable the RX half so you can't get in. I've never actually checked.
How about Southern California? I use to listen to the re-broadcast on the JPL repeater, but haven't noticed it rebroadcasting over the past few years. Perhaps because there weren't any JPL projects on board? Of course there is always NASA TV!