There is a small supermarket on site and also a spar like shop opposite the entrance to Sea Club.

We found a large supermarket behind (about a 10 min walk) the sea club - that was fab - couldn't tell you address or it's name but Im sure one of the reps or somebody on reception could direct you.

The two cheapest (national) supermarkets closest to Sea Club are Eroski Syp on the main road going away from Puerto Alcúdia, about 20 minutes walk. (There is another Eroski going in the opposite direction into the port about the same distance.) Caprabo is on the Avenue Tucan pretty much opposite Hidropark (probably the one Hippyjak's referring to). I'd say it was further than 10 minutes, but I'm hopeless at judging this!

Tourist supermarkets are way more expensive, though some of them have much better ranges of booze than the main supermarkets and at better prices.