When designing one of the FTDI USB bridge chips into my product, can I power the VCCIO power rail from an external power supply and run VCC off of the USB VBUS?

FTDI recommends that the entire chip be powered from voltages derived from the same source.  If the USB VBUS is intended to power VCC, then the on-chip generated 3V3OUT or another voltage power converter with USB VBUS as the source should be used.  Powering only one “side” of the IC will result in the chip being in an unknown state.