The other power supply to the AC switch comes from the same fuse that powers the blower motor relay, so that fuse must be OK if the blower works. The switch shares it's ground with the blower motor too, so a ground problem is unlikely if the blower works, which kinda suggests the switch itself is faulty. A bypass test of the switch would confirm but it's not that easy to get to the switch harness connector to do it.
There is also a small resistor contact on the Fan Speed Control in the Environmental Control Panel - The Fan Speed Switch). If this resistor is broken or not functioning the AC will only work in low fan speed one, maybe two - but, if your fan speed is set higher the AC will not work. My wife usually just pushes the fan speed witch in hard to clear this problem. But that is a short fix (don't know who told her that trick - but it isn't a fix).
I believe the resistor is meant to provide a signal to the AC system that the fan blower is ruinnig fast and to boost the AC rate (don't flame me if this is not correct - it is my observation. Some one at Hyundai may be able to shed light here, but eb=very time I ask about it they simply tell me the resisteor is a complicated function and offer no explanation)
Incidentally,,,,, I noticed this when I had to reassemble my Radio/AC Control Panel after some tweeker broke into my wife's car and ripped the benzell off to rip out her fancy new Radio she had just had installed. When I got it all back together the AC did not work. So, thinking I had done something wrong, I took it apart again and could not find anything amis, except a small little wire that was loose and had no where to be connected.
So I went to junk yard and tool one apart in the junk yard. I carefully analyzed the disassembly and looked for that little wire. When I found it I noticed it was fragile and connected to the Fan Switch. So I took out the fan switch and bought it.
When I got home I took her car apart again and removed the fan switch. Putting them side by side I found a teeny tiny contact on her fan switch where that little wire attached to. I mean it was the size of maybe two sewing needles in width. Anyway, I installed the fan speed switch from the junkyard and reassembled her car. THE AC WORKED!!!!