Is there a rule where the opponent scores if you cycle through your deck?

Yes, there is a rule regarding this. When you need to draw a card or move cards from your Main Deck to another zone (like the trash) and your deck is empty, you perform an action called Burning Out.

According to Rule 431.2, when you Burn Out, you must:

1. Perform as much of the action as possible.
2. Recycle your trash into your Main Deck (randomizing it).
3. Choose an opponent to gain 1 point.
4. Complete the remainder of the action.

Additionally, per RiftJudge FAQ #6736, the point gained by an opponent when you run out of cards can win them the game, even if they already have 7 points, because this point is not gained through the "Conquer" mechanic (which has stricter winning requirements).