Can a Variable Frequency Drive convert standard wall outlet (120V/15A/single phase) to ~220V 3 phase?
There are VFDs that will accept 115V single phase input and give you a 230V 3 phase output. They use what is called a "voltage doubler" front-end rectifier, meaning they rectify the 115VAC to approx. 165VDC, then put it through a circuit that doubles the DC to 330VDC, then the inverter section uses PWM to recreates an output that MOTORS react to as if it is 230VAC 3 phase. The key point however is MOTORS; this can ONLY be used for 3 phase AC induction motors, you cannot use the output as a general 3 phase power supply source. it is the inductive nature of the motor that allows the PWM to become a pseudo-sine wave. Without the motor, it's just pulsating DC that changes direction.

Then there is the size issue. On a 15A 115V circuit, you can actually only load it to 80% of that continuously, so 12A. The largest 115V input VFD is for a 1-1/2HP motor, but the INPUT CURRENT on the 115VAC side is actually 20A; too much for a standard 15A circuit and actually, too much for even a 20A circuit (80% rule means a 20A circuit is only good for 16A continuous). For that reason, MOST of the companies that sell a 115V input VFD limit it to 1HP, and even those require a 20A circuit (the NEC allows you to use a 20A breaker for a 15A outlet if you use 12ga wire).

Bottom line, if you have only a 15A 115V outlet, you cannot (legally) connect a device to it that requires 1.5kVA continuously, that is too much for that circuit regardless of what you put on it.