If I as the stepgrandparent and my husband (biological grandparent) share custody of my stepgranchild with the child's natural parents is the grandchild eligible for benefits under my (stepgrandparent) record?

Hi,

A grandchild or step-grandchild can only become eligible for child's benefits on a grandparent's record if they are adopted by their grandparent, or if both of their natural or adoptive parents are either deceased or disabled (https://secure.ssa.gov/apps10/poms.nsf/lnx/0200306235).

Best, Jerry