How is difficulty defined? Is the card being declined when used? Or is the merchant saying they do not want to take the card?
Beyond that are these large purchases or small? What brand of card (If Discover for example, then yeah, big problem, Visa? Should be no issue).
Also, they probably should call their card company and check if a hold has been placed on the card.
I can say, it has nothing to do with politics, or the fact that they are out of the city. Paying by card is common all over Portugal and the Algarve is a heavy tourist zone, not a remote mountain village.
Used my card in Lisbon and the Algarve (Tavira) last year, with the exception of a few restaurants that took only cash, and some small purchases where I did not even bother, I had no issues.