Been sailing the Virgins for 37 years. Kept my boat there for 11 years, until 3 years ago. I agree with much of the advice above re: the BVI being the ideal place for a newbie charterer.
The suggestion to reschedule your charter almost anytime except the Christmas holidays is a good one, both because of the crowding (which isn't really a huge problem; you can still anchor in a few places, but the moorings have multiplied like rabbits and they're still often all gone by early afternoon)..and the potential problem of the so-called "Christmas Winds". These don't come every year, and when they do come they usually blow for only a few days. However, two BIG caveats:
1. Christmas winds are not just stronger-than-normal tradewinds; they are mini northers which can blow like stink....40-50 knots is not uncommon; and
2. Christmas winds rarely but sometimes blow for many weeks, as they did in 1992-3.
Bottom line: if your only time window is Christmas, by all means go, but be prepared for crowded anchorages and possible weather unpleasantries.
Re: companies, there are lots. Moorings is top notch, but also expensive. Sunsail is probably next in terms of maintenance, experience, etc. The Moorings spinoff, Footloose, is a possibility. They use boats from the Moorings which have been moved out of service, and are cheaper. Heard good things about Horizon. Heard very few good things about North-South.
Be flexible. Have fun!
S/V Born Free