Not sure from your post if you are trying to book it diy e.g. hotel and flights separately. If not, then I think that's your best chance.
I fly to majorca regularly as my boyfriend lives there, so you shouldn't have any problems getting flights (I usually pick them via skyscanner.net). I often fly with one airline and return with another.
If you look on the Majorca forum, there are lots of posts about accommodation so I would book that first, and then book the flights to tie in with whatever dates you get accommodation for. You may find a private let would be better than a hotel (although they may sleep more than 2), but it will depend which resort they want and what other facilities.
Unfortunately a minority of teenagers these days are giving the majority a bad name, so hotels may be more inclined to set an 18 limit as they are legally classed as an adult (although they have no doubt been paying adult prices from about 12 or 14 years old!!)
Hope you find something soon.
Louise