Some VFDs yes, other VFDs no. You'd best provide a specific VFD model. In general, typically, I think the answer is "yes".
Lol. like briselec below, I thought OP meant Variable Frequency Drive as well (just like Google).
A vacuum fluorescent display? A specific VFD requires a certain amount of DC filament voltage, about 60V DC from memory? It might be interesting to somehow rectify and chop down mains electricity to get your 60V DC, but if you did that in the naive way you'd end up with the 60V DC with respect to mains neutral, instead of mains earth. This is very dangerous. You need to use an isolated transformer for safety.
Given that you need 5V or so for whatever circuitry drives the VFD, I'd probably suggest that you just step up 5V (or 12V, or whatever else you have handy) up to the 60V DC. It'll be less complex than creating a safe supply from mains.
Do you have a particular design in mind for how you're going to convert 120V AC to ~60V DC? How are you going to power the microcontroller (or whatever) that will be driving the VFD?