The first thing to check is if the fuse or relay is blown. Anyone can check is to see if the AC fuse or relay is bad. It is what you should do first before paying a dealer or a shop labor for a diagnosis. Your owners manual will tell you where they are located. Some vehicles will have a fuse and a relay. The relay being there to protect the electrical system when the clutch engages, which causes quite a draw of amps.
AC compressors since the mid '80's are designed not to engage the clutch if it is low on freon. They will cycle on and off if they are pretty low, say with about 1-2 lbs of freon. This is so the compressor will not overheat and get damaged. Unless you know how to work on the AC system, I would have pro's diagnose what is wrong. If the system is just very low on freon, you would want to find out why. So, if it is just a leak somewhere, you would want to get that solved before refilling it with freon. A can of freon dye and a UV light will show exactly where it is leaking. If the clutch on the compressor is engaging and running, then you may have an HVAC door operation problem in the AC box. It could be stuck in the heater mode. This could be caused by a bad servo in the AC box, or the AC controller in the dash not functioning correctly.
As others have said, putting the HVAC system on vent doesn't mean you would get ambient air temps through the vents. The air could be quite a bit hotter has the air is drawn in at the base of the windshield, right behind a hot engine. Hot under hood air temps could be going right into the cowl vents.