1095 A form If my children receive insurance through the marketplace, do I put the 1095A form information in on their tax return on mine?

If you are claiming them as a dependent on your return, then the 1095-A goes on your return.

If they are not claimed as a dependent on any return, then the 1095-A goes on their return. 

Whoever claims them enters the 1095-A on their return.