Question

would it be grammatically correct to say want in this context::::
As the child grows so does it’s want for interaction with others.

Answer 1

I would say need instead, as it conveys the idea better in my opinion, since "want" can be so subjective.

Answer 2

Want in the above context means to have a desire for. So reworded, the sentence would be:
As the child grows so does it’s desire for interaction with others.

So to answer your question, yes it is grammatically correct.

Substituting want for the word need as *Sumiko* suggests will change the meaning of the sentence. Using need in that context means require. So the reworded sentence would be:
As the child grows so does it’s requirement for interaction with others.

So

As the child grows so does it’s want for interaction with others.
As the child grows so does it’s need for interaction with others.

are both grammatically correct, sort of (more on that in a second); it just depends on what you are trying to say. Does the child require the interaction or does the child desire the interaction? If you are having a hard time answering this question, think of it this way:

If you say the child needs or requires the interaction, you must be basing this statement on another premise or argument such as a study that shows that children deprived of this interaction become mentally underdeveloped or something, i.e., child development.

If you say that the child wants or desires the interaction, you must be basing the statement on another premise or argument such as a study that shows children gravitate towards others more as they grow, i.e. child behavior.

So in essence, is your paragraph that this sentence is in discussing child development or child behavior? Choose the word that best fits.

Now while the use of the word want in the sentence is just fine, the use of the word "it's" is not grammatically correct. "It's" is always always always a contraction for "it is". There is never any exception to this rule. When you are using the pronoun "it" and want to express possession you spell it "its". If you can remember that "it's" is a contraction, you will always know when to use it's or its because you can just ask yourself, "does it is fit?" Look at your sentence as an example:
As the child grows so does it’s [it is] want for interaction with others.
See that doesn't work.

There is one other grammar error in the sentence; you forgot yur comma.
As the child grows
is a dependent clause because it is an incomplete thought (what happens as the child grows). As a dependent clause, it needs to have a comma following it to show that the clause that follows it is completing the thought.