and then use her debit/credit card (as long as it has the Visa symbol on it) to pay for it.

I think renting hotels with cash went out a long time ago as there is no one responsible if you trash the room and then leave it, etc.

Just do a simple search and use keywords. From there, you will have plenty of resources of the majority of hotels within Arlington. It also provides their address and phone number. Just call them and ask. It does not sound weird, don't worry. I've done it plenty of times. The staff knows that some people do not have credit cards or use credit cards, and that is the reason to why they ask the question. So, it is not inappropriate to ask.

There are plenty of hotels in Arlington that will use cash only, and does not require a credit card deposit of any kind. Hope that helps.