In the state of Virginia, lodging an appeal requires you to show that you have the necessary funds with which to pay the damages cost, and in appealing the case to another court you have to show that you had intent to pay the damages awarded.
She can't claim bankruptcy, that's not going to work. She could borrow the money from someone, but she already owes $6m in legal fees.
Depp's lawyers have already implied that they are not going to make her pony up the money or garnish her wages. If she wants to appeal the case, Amber Heard has to come up with the money to pay the damages with interest in case she loses the appeal and is ordered to pay over even more money.