Can I make risotto without wine?
Yes, all it is, is flavour.

Apple juice and grape juice are 2 things I've used in the past for non-alcohol people. You can also get away with not using anything as a replacement as long as you use enough of everything else (butter and Parmesan etc). 

Wine just gives a little depth and a sort of 'freshness'.



Edit:

I just read the vegetarian part of your question. I'm assuming you already know, but Parmesan isn't vegetarian, as it is made from rennet (rennet is extracted from the lining of the inside of the stomach of mammals), and most commonly from the fourth stomach of young calves. The vegetarian substitutes I've tried in the past (Called pasta cheese), are not great. However, unlike Parmesan, Gruyere is allowed to be made with vegetable rennet and tastes great (I personally prefer it over Parmesan).