I've had corporate travel services place tickets on a 24 hour hold for me before (i.e. the itinerary is reserved, but not yet paid for). I assume that's still possible, at least for the legacy carriers.
Continental used to have that, but dropped it shortly before the United merger. I think other companies did too. Instead, they let you cancel with no cancellation fee within 24 hours. (Although the funds may take up to two weeks to go back to your account).
So essentially, FancyHands can book it for you with their card, and cancel it if they can't charge you within 24 hours - but I wouldn't be surprised if they don't want to go that route.
American Airlines still has this feature (or at least it did last night when I was making a booking)
Edit: By this feature, I mean the ability to place a hold without the credit card.
So essentially, FancyHands can book it for you with their card, and cancel it if they can't charge you within 24 hours - but I wouldn't be surprised if they don't want to go that route.