I make a vegetarian matzo ball soup every year at Passover in addition to the regular kind with chicken broth & fat. My son is a vegetarian so he eats the veg version while the rest of us eat the traditional one. It's easy. I just use vegetable broth instead of chicken broth. You can buy vegetable boullion cubes at the grocery store, Knorr makes them, or vegetable soup base from Penzey's. Then just use olive oil or vegetable oil instead of chicken fat in the matzo balls.
No, it doesn't taste identical to the chicken kind. But it's still pretty good. Especially if you throw in a bunch of chunks of vegetables into the soup to give it some oomph.
Not that the OP should have accommodated her vegetarian friend's request. I agree that it was ok for friend to ask once but then she should have taken "no" for a answer and made her own veg soup if she really wanted that. I'm just saying that vegetarian matzo ball soup is not only possible, it can be quite good.