Q:

Is there an ATM at or near Vágar Airport?

When we arrive at Vágar Airport (FAE), will there be an ATM in the airport? (If not), what about the nearby town of Sørvágur?

A:

Two are shown at https://www.mastercard.us/en-us/consumers/get-support/locate-an-atm.html
A DuckDuckGo search for ATM locator shows many options.  Above is the second hit.  First hit was Visa and it said there were none in 600 kilometers of either of your locations.
You can also see them on Open Street Map, or if you want a list, see https://stackoverflow.com/questions/23684689/open-street-maps-query-for-atm

