Most maki sushi (rolls) contains nori (seaweed) which will give it a "fishy" taste, even if the roll only contains vegetables and no fish.
Some restaurants have a roll or two that are wrapped in a thin omlette or cucumber instead and don't usually have nori.
It is not good form to ask for them to leave out the nori - some chefs may be offended by this.
Most places also serve nigiri sushi, which is a small mound of rice with a piece of fish on top. I have never seen veggie nigiri, but many places have nigiri with egg on top.
Last but not least there is inarisushi, which is a tofu skin pouch filled with rice (and sometimes veggies). I don't like it, but it has neither fish nor nori.
Honestly, if you are grossed out by fish or the taste of nori you probably shouldn't order any sushi. Most sushi places (even the conveyor belt ones) offer other food options such as tempura (battered and deep fried veggies or shrimp), fried tofu, chicken or pork katsu and sometimes terriyaki, udon and other warm items.
Good sushi fish btw. does not taste fishy. Most fish will only taste or smell fishy when it is not fresh (with the exception of some types of fish such as mackerel).
A good fish to start out with, if you really want to try sushi, is salmon or tuna.
Good luck.