Salmon oil is an excellent source of Omega-3, one of the essential fatty acids that
supports the immune system and promotes health of the heart.
A) supports the immune system and promotes -> "that" modifies "fatty acids", which is plural. So, we need plural verb, not singular verb, supports. Incorrect.
B) support the immune system as well as promote -> SVA error is corrected. But, "support the immune system" and "promote health of the heart" are different items. It will be better, if we use "and" as conjunction. Incorrect.
C) support the immune system and is a promoter of -> We lost the parallelism. The items "support the immune system and is a promoter of" are not parallel. Incorrect.
D) support the immune system and promote -> It is better. Let's keep it.
E) support the immune system and are the promoters of increased -> We have the Parallelism error again. Incorrect.
So, I think D.