My ex and I have two children together. We were never married and do not have any legal custody agreement. Can I move the kids?

No you can't move without his written permission or permission of the Court. You also need a custody judgment. Contact a family law attorney for both. Also consider child support. To move out of state, a judge is going to want evidence and testimony that the move is in the kids' best interests.

You would have to comply with the relocation statutes in order to relocate out of state with the children.

Louisiana law requires that you give him notice of your intent to relocate by certified mail including why you are relocating, the address, the school the child will be in, and why the relocation is in the child's best interest.
If the other parent does not oppose the relocation, be sure to get it in writing. If the other parent opposes the relocation, you will need an attorney to file a petition with the court to address paternity, custody, relocation, and other incidental matters such as child support.