Let's say I have no cards left in my deck when my opponent's turn ends, the "Lost World" Stadium is in play and my opponent has 6 Pokemon cards in the Lost Zone. Can I declare myself the winner at the beginning of my turn, before drawing a card from my deck?

No you can't. You can only declare yourself the winner with the effect of Lost World during your turn, but since you can't draw a card from your Deck you cannot start your turn, and therefore you lose.

Relevant cards