He is deeply concerned for vulnerable children.
Is this correct? Or is there a better preposition to use instead of for? What exactly does this phrase mean?
What about "He is deeply concerned with vulnerable children"?
There is a few variation to be used with to be concerned.
One can be concerned with something. This would have a meaning similar of to deal with
I am concerned with daily maintenance of our machines.
One can be concerned about something/someone. This would be the preposition which could be use in your case, but it would give it a meaning of caring about.
He is deeply concerned about vulnerable children
Of course, for seems to be used, although I rarely heard it (which is in no way an indication of the frequency it could be found), it is used.
EDIT: See Dan Ray's answer for a use with the for preposition.