What can she do to get him to pay child support across the state?

The child's mother needs to file an action to establish the parental relationship between the mother, the father and the baby in the Oregon county where the baby was born. That action would also seek to establish a parenting time schedule for the father which would be one of the factors in determining the appropriate amount of child support he would be ordered to pay. He would also have to get served with those papers. Once there is a judgment in place in Oregon, the other state where he lives is under a legal obligation to enforce and collect the child support (as Oregon would be for a child support order from that state).