Honestly, an aircompressor doesn't need a VFD.
If I had an aircompressor that had a 3 phase motor on it, and only had single phase power, I'd put a single phase motor on it. I really don't see much point in trying to run a VFD on a compressor, unless you were going to put an air demand based speed control on it to save power, or make more airflow. Or use the VFD to overspeed the motor at low tank pressures if the compressor head is rated for the RPM. For just your standard garage air compressor, I personally think it's better to stick a single phase motor on it. Unless you allready have an RPC that can run your compressor, and whatever other machine at the same time. Then I'd run it off the RPC.
With a VFD you can do some NEAT things with a compressor. You can set it up so the relay inside the VFD switches an air solonoid valve as an unloader so that when the motor hits it's target speed, it closes. This allows you to get rid of the unloader, and have a LOT LESS inrush current that might trip breakers. You could also overspeed the motor IF the compressor head itself can handle it, and have it slow down to normal speed as the pressure builds. Like say set the motor to 80 or 90 hz when it hits the starting pressure, and have it slow down to 60 when it gets near shutoff. You'd have to build your own pressure sensor, but that's not that hard. Some compressor heads come with smaller motors, and spin them at slower speeds on different model compressors. You also have the advantage of having as much time as you need to get the motor up to speed before the unloader closes. In other words, you can get a LOT more airflow out of a smaller motor with a VFD. If I were going to use a VFD on an aircompressor, I'd take FULL advantage of this. You can also program different modes. Say you want to give the compressor less abuse when the airflow isn't needed. You can set it to run slower when you only need minimal airflow. Or you can have a max airflow mode where you overspeed the motor when the pressure drops. You can also put a temperature sensor on it so that if it does get too hot, it slows down, or shuts off. You could also put a knob on it to make the airflow variable. A VFD could make a compressor a LOT more efficient, and effective.
At the very least, you'll probably have to put an electric unloader on it, because I dunno if the motor will be able to acelerate fast enough to not cause an overcurrent trip without it. The VFD has a relay output you can use as an electronic unloader. You'll also have to wire the pressure switch up the the 24v terminals on the VFD. It will work just doing those 2 things. DO NOT use the pressure switch to switch power to the VFD or the motor. You can try it without an electronic unloader first if you like, but I wouldn't count on that working.