Well I'm sure that most vegetarians don't need to take seaweed or kelp suppliments. I'll bet that the vast majority of the vegetarians on this web site don't. The only nutrient I'm aware of that exists in sea plants and doesn't exist in other vegetarian foods is EPA. EPA is a long chain omega-3 fatty acid. It's found in fish, certain marine micro algaes and wakame (which is a type of seaweed).
DHA is another long chain omega-3 fatty acid that our bodies need. It's found in fish, certain marine micro algaes and eggs from hens that were feed flax seeds.
Now our bodies are supposed to make it's own EPA and DHA given the right conditions. Two of the most important conditions are we need to consume enough omega-3 parent fatty acids (LNA) and not too much omega-6 parent fatty acids (LA).
Well, if you want to make sure you're getting the DHA and EPA and not rely on your body to create them you can get them through direct sources. As I said, wakame is a source of EPA. You have to eat a considerable (but realistic) amount though. There is 186mg EPA per 100g fresh wakame. There are micro algae EPA capsules on the market but none of the ones I've seen today are vegetarian because they contain gelatine. That might change though. Also our bodies can convert DHA to EPA.
The direct vegetarian sources of DHA are eggs and micro algaes. There are vegetartian micro algae DHA capsules on the market.
The omega-3 and omega-6 levels of vegetarians and especially vegans are far from optimal (generally speaking). So it is reasonable to consider taking direct sources of EPA and DHA. But, for most people it's not essential if they keep the LNA and LA consumption at reasonable levels. LNA is the omega-3 in flax seeds and LA is the omega-6 you see in many nuts and oils.
Sorry for this being so long winded.