I know there are many, MANY bands that play Disney songs at live events, and never once have I heard of Disney so much as saying "boo" about it. Many bands also use Disney tunes on their CDs - and I believe most of them have purchased the right to use the music on a predetermined number of CD printings. Generally speaking, recorded music or broadcast music (such as my radio show) are held under tighter scrutiny than live performance or local cast (such as a wedding DJ.) Ethically speaking, I think that's appropriate - live and local usage of a copyrighted tune hardly measures the same as mass distribution, and by and large many copyright holders seem to agree. What the law states, however, is often a different matter, and it only takes one ass-hat to ruin your day with a lawsuit. I've very occasionally heard of copyright holders going after live performers for using a song, although usually those are big-ticket artists who've never written a pirate tune.
Most pirate/shanty artists are very accommodating, though, and largely happy to let you play their music at events. So ask and see what happens. Just be careful about misinterpreting permission to play an occasional live song as permission to record it, and also be careful about "traditional" shanties that actually aren't (there are several songs that folks assume have been around since the 1800's that are actually relatively new, and still under active copyright.)
Oh, and just to be on the safe side, I am not a copyright lawyer, so please disregard everything I've just said.