Question

The children on the baseball team. is that a complete sentence

Answer 1

No it is not a complete sentence. You are missing another clause - that is a simple clause not a sentence:)

You need something further on, or a connective or time connective before the subject ... "The children".

Hope I helped:)

Answer 2

No "the children" is the subject and "on the baseball team" is a prepisitional phrase. It is not a complete sentance because it is missing the verb.