Can the other parent refuse visitation because I am behind on child support?

The short answer is No. Your duty to pay child support is not at all related to the other parent’s duty to surrender the children according to the order in place. You both may be in violation of your orders. You should appear at the appropriate place to collect the children for visitation. Make notes on a calendar. Pay your child support; because when you take the other parent to court over lost time with your kids, you will have to address the child support as well.