Is Childrens' a correct plural possessive form?

No. The plural possessive would be children's.

'Children' is already a plural quantity. It is the plural noun for 'child', for any number more than just one child. If something belongs to any quantity of children above just the one, then since the noun 'children' already denotes a plural, from two to several, you do not add an 's' to make what would be a double plural. So, you have only one 's' needed: to describe ownership & possession, and, as is usual, the apostrophe comes before it.