1. Cheese Pizza from a Non-Kosher Store
As long as the store confirms that the pizza is fully vegetarian — meaning no meat products, no rennet in the cheese, and the cheese is microbial or otherwise vegetarian-friendly — then yes, you can eat cheese pizza from a non-kosher store. The main concern is making sure that there’s no mixing with non-kosher ingredients and that the cheese doesn’t contain animal-based rennet. The fact that there was a previous pizza in the same oven (or even at the same time) with peperoni, does not pose a problem, because there won’t be flavor from that peperoni on your pizza.

2. Vegetable Toppings (e.g., onions, tomatoes, etc.)
Vegetable toppings are generally fine as long as they’re just vegetables and not cooked or mixed with anything non-kosher (like meat or meat sauces). It’s always a good idea to ask how they’re prepared (if not raw), just to be sure.

3. Onion Rings and Fries
These can be a potential issue if they are fried in the same oil as non-kosher items, like chicken, meat, or seafood. However, many places (especially large chains like Domino’s) use dedicated fryers for their fries and onion rings. You can simply ask them if that’s the case. If the oil is separate and not shared with non-kosher items, they’re fine to eat.

4. If a Kosher Pizza Store is Nearby
Even if a certified kosher pizza place is somewhat nearby, halachically you’re not required to go out of your way if you’re able to confirm that the vegetarian pizza at the closer, non-kosher store meets kosher requirements. Convenience is a valid consideration as long as the food itself is kosher according to the guidelines above. When something like this is kosher, it is kosher, and not only to be used in dire circumstances.

So in short — yes, as long as you verify that the pizza and toppings are fully vegetarian, and check on the fryer situation for fries or onion rings, it’s fine.