Can I say "The organization contributes for children in developing countries" Can the word contribute have someone as objective?
Depending on your meaning, probably not. The sentence you've provided is unclear.
If the organization is contributing money to children in developing countries, you would use to.
The organization contributes to children in developing countries.
The use of for is ambiguous, because it can mean one of two things:
The organization contributes for children
In this case it can mean the same as the above to sentence, or it can also mean on behalf of the children. So perhaps the children gave the money to the organization and asked the organization to give the money to someone.
I was unwell, so my brother posted the letter for me.
In this case it means he posted the letter (on my behalf).
There was a cake addressed to her, and a cake addressed to me. First, she opened the cake for me. Then she opened her own cake.
In this case, she opened the cake that was intended to be given to me. She didn't open it on my behalf (for example, because I was not strong enough to open it).
Because of the ambiguity it doesn't sound very natural to a native speaker to use for in this case, and you should use to.