In a Las Vegas hotel like the Bellagio, can you charge shopping to the room?
I am going to las vegas soon and do not want to carry cash around with me as I shop. Is it possible to just charge things to the room and have it go on to that credit card instead?

The only thing you can do that with is like room service, dry cleaning or anything that the hotel itself provides. The stores on the property only rent the space from the management company and do not allow payment through your room.
hotel run stores only will let you charge back to your room.
You can charge almost anything back to your room from any MGM Mirage property.
You can buy clothes at MGM Grand, have dinner at Mandalay Bay, tickets to LOVE at Mirage, and bottle service at LAX and sign it all to your room at Bellagio.