Does anyone know, if you are accepted into an LLM program in the US and don't have a scholarship, from your own country, will the school likely give you one, if it means you cannot otherwise attend and will have to give up your position. Can you at that point ask for any additional scholarships/grants?

They can give you a scholarship if you fit the requirements to receive it (e.g. academic merits, financial need, etc.), but its difficult to say if that will "likely" happen or not. It depends on the university and your profile. You should mention them your financial problems and ask them how is the procedure to apply for a scholarship.

One thing is for sure: no one will give you a scholarship just because it happens that you cannot attend and you would be forced to give up your position... they can give your position to another students from the waiting list that is willing to pay. But if you fit the profile, they might be willing to help you even if you cannot afford it.