If my opponent is using a coin as their randomizer and it seems suspicious, do I have the right to ask them to use a die or call over a judge about it?
Head judge has final say in whether a randomizer is legal or not. There is not anything wrong with calling a judge to be sure.
Just be aware, currency (as well as RS-on theme deck coins) is a form of legal randomizer. Only call a judge if you feel there's actually something fishy.
The rulebook specifically says "Any coin released with any Pokémon TCG product from EX Ruby & Sapphire on should be considered fair and impartial. Any other coin (such as local currency) brought by a player to be used as a randomizer must be approved by both players."