Is there an ATM at or near Vágar Airport?

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

There used to be an ATM at the airport, but I am actually not sure if there still is one in the new terminal :-) See: https://www.youtube.com/watch?v=MsKqPSE65RI

But there are ATMs both in Sørvágur and Miðvágur near the airport.

And if you are traveling to Tórshavn with bus, there is one at the busterminal.

See also: http://www.visitfaroeislands.com/plan-your-stay/before-you-arrive-in-the-faroe-islands/money/