My ex boyfriend and I share joint custody of our son. He moved across the country, and hasn't seen our son in 5 years, but he's been paying child support. There is no order in place to determine who claims our son. I assumed if he lives with me and his father has no contact with him, that I get to claim him. His father said that he intends to this year. Can he do that?

No since he is not a custodial parent but you are; you have a child but you're not married, then, as in your case, you can end up fighting over who gets to claim the kid as a dependent. The custodial parent, you, usually get to claim the child as a dependent for the year. The IRS defines the custodial parent as the parent with whom the child spends the most nights. You count the child as spending the night with you if he either sleeps at your house or spends the night with you.