Kelp has many trace nutrients and mineral in it. It is good for you but you don't necessarily have to have kelp supplements.
Protein balance is probably the one thing you need to be aware of. Soy products will provide protein. Are you vegan or lacto vegetarian or lacto ovo vegetarian? If you still incorporate dairy products you can pair up a dairy item with a whole grain item and get a full complement of protein. Eating eggs (ovo vegetarian) will give a full protein. Whatever diet style you follow is fine. Just be sure your protein is adequate.
Eating a vegetarian diet should provide the nutrients you need. I am not sure why you were told to get a supplement for sea weed. It won't hurt but I am not sure it is necessary.