The SHO has a Nippondenso 10P15F compressor. It is made to take the high RPMs of the SHO, and is only a few dollars more than other Ford vehicle compressors.
That having been said, Steve is correct. Do not start replacing parts without first doing the diagnostics. You can jumper the connection to the low pressure switch to see if the compressor will engage. If the compressor engages, then either the system is low on refrigerant charge, or the switch is bad.
If the compressor fails to engage, then it could be the clutch coil, which is replacable as a seperate unit, or there is a failure in the wires or connections of the A/C system.
There is a great article put together by the Utah SHO club and it has been posted numerous times on this forum. It walks you through the tests of the A/C electrical system, including the fuse, relay, wires, connections, and switches. Start your diagnostics there. Use the search feature.
Once you have a better idea as to what the problem is, or is not, then we can move toward solving it with the least amount of expense.
Look for black oily dirt around your A/C connections, behind the A/C clutch coil, on the front of the condenser (in front of the radiator). This is the sign of a leak. If your system is low on charge, then it has a leak, period. If you find the signs of a leak, then it is a good bet that that is why your compressor will not engage (low pressure due to leaked out refrigerant)
But, do the diagnostics suggested above so you can rule out other problems, AND become familiar with your A/C system. rangerj