At work today I wanted to tell our guests that a car was waiting for them. Is there a formal way to say that? Are "The car is waiting for you" and "The car is ready" correct?
A formal way to say this would be "The car is now ready at your convenience."
If the car comes with a driver, you might say "Your driver is ready at your convenience." It is implied that the car is also ready (or the driver wouldn't be).
"At your convenience" means "at a time suitable for you." It is a very respectful way to let your guests know that they are not being rushed and that they are being cared for and/or provided a service.
If you want to avoid the suggestion that the car is 'waiting' (which could be seen as impatience) or 'ready' (less intense, but same possible misread connotation), you could simply say the car 'has arrived', and if you want to be formal AND helpful, tell them where it has arrived (at the front of the hotel, in the back of the hotel, at the east/west wing, which you can get to by...).
By saying it has 'arrived', you leave no weight upon your guest's shoulders (a waiting car could lead to the expectation that they should hurry), and more importantly, by telling them where the car is in relation to their own location, you are being very helpful, which the guests are likely to appreciate.
"Your car/taxi is outside" is better. Then it doesn't feel like you are rushing them.