I recently bought a fair few Prize Bonds. The first purchase for yourself needs to be made in an Irish post office. You can only purchase online if you have a pre-existing online account, which is created by the application itself. They need to verify identity, address etc. by copying a few documents in the post office.
Prize bonds can be bought as a gift for you by other people, so if you know someone who already has an account they can purchase the first tranche for you.