Quick disclaimer, I've never played this game before but as the rules are only 6 pages long it was worth reading rules and seeing if I could find the answer.
Looking at the rules here I cant find any references to having to give players resources from your own pile. There doesn't appear to be any stealing like in Normal Catan.
Instead it looks like you might have misinterpreted the following on page 4.
Because you rolled the die, you must give your fellow players
any newly earned resources from the stockpile.
Look at the picture below. You are the white player and roll a
“4.” You and the red player each have a pirates’ lair next to the
sugar cane island with a “4,” so you each get a molasses tile.
A treasure cave island is also marked with a “4,” so you must
give the blue and the orange pirates 1 cutlass tile each.
The stockpile appears to be just a general supply by the board (as described on page 2 under 'resource tiles'. The way these rules are wrote its implying that the player rolling the die is responsible for giving other players resources they are entitled to. I believe the line "must give the blue and orange pirates" means give them from the stock pile not your own pile. I think if it was your own pile that would be more explicitly stated and all new resources always come from stock pile.
So in answer to the question what happens if a player cant give a resource they need to I would say that is an impossible situation that should never happen.