In the state of Oregon, How long must a custodial parent wait before they can get remarried and does the non-custodial parent have any say in this?

You can remarry as soon as the judge signs your divorce decree.  Unless you are marrying someone with a criminal history or some other issue that would impact custody, remarriage should not be a negative factor in a future custody decision.  No, your ex has no say in when you can remarry.