It would be possible, but I think it would be hard. Eating and cooking would be a full-time job, as your normal sources of protein have been incredibly limited. With all that, you probably wouldn't be able to eat soy either. That would be tough.
There's always nuts! Though I imagine you would have to eat a lot of them. Your diet would be veg, nuts and potatoes. Sounds dreadful to me, but I already have a dairy and soy sensitivity.
It sounds pretty tough. Eggs, dairy, and beans are great sources of vegetarian protein, and you'd be giving up all of those (presumably soy as well since that is a type of bean?). And being allergic to gluten AND corn AND rice? Yikes. Good luck!
I'm sure there's a way to live as a vegetarian with those allergies, but I'm not sure how. I was a vegetarian for two years, and my mom has been one for four. We eat a lot of beans, rice, eggs, bread, and pasta. If you wanted to try the vegetarian lifestyle, I would recommend talking to a nutritionist.
Corn, rice and gluten all at the same time? That guy does not exist. YET. But the guy dont necessarily have to be vegetarian. You can still consume seafood, honey, lactose-free yogurt and milk, eggs etc.
Eating habits are not innate or permanent (save for animals). It is said "do as the Romans do", so if one has to shift from dairy to veg, no problem. What will one do if he/she doesn't find food of choice? Obviously change!
I think itd be easy. You could eat eggs. Or just become a vegan right away, better for the earth ;D Im a vegetarian planning to be a vegan in like a year, probably right after when i move out of my parents house.