ATMs in Appenzell?

In theory, you can go to Google Maps and search for ATMs in Appenzell. There seem to be a few, but I cannot guarantee the information is 100% up to date. You could email your lodging or the manager of your lodging and ask. They will know.

I travel a great deal in Europe and have only twice had an issue finding an ATM when I needed one. The first occasion was in a small town on the Cornish Coast, pre-pandemic. For budgetary reasons, a lot of banks in the UK had been closing branches, and Mevagissey had no ATM at the moment. The local residents were trying to resolve the problem. Mevagissey's about 1/8 the size of Appenzell.

The second problem was post-pandemic in Oslo, of all places. Norway is a country where almost all your business can be transacted by credit card; plenty of places have posted signs stating they do not take cash. Trying to find a bank-owned ATM in Oslo was a real challenge. Most of the bank branches I found on Google Maps had no ATM; most didn't even have a street-level office. The one bank ATM I eventually located had a line of about a dozen people waiting to use it.

I have no reason to think Appenzell is going to resemble Mevagissey or Oslo.