Minesweeper is the best puzzle to be made in physical form in the style of a scratch ticket. (Not good for the actual lottery; people will repaint incorrectly scratched mines if money is riding on it.) Perhaps it can be included as a fun irrelevant puzzle in lottery tickets.
What other puzzles would work?
Sudoku as scratch ticket might be a good way to temporarily conceal then deliver the solution. It could include "mines" of squares that are not solvable by logic, though that might be a tricky thing to define. Perhaps a sudoku that does not have a unique solution.