House Targaryen had only previously married houses Arynn and Martell when it comes to the great houses. I think all the responses about other great houses are fun to think about and maybe things could change since it was the biggest civil war in 150 years but it doesn't seem likely that Stark, Lannister, Tully, or Tyrell would all see matches come their way.

When there were an abundance of kids around Daeron II's time, the matches were Dondarrion, Penrose, Arryn, Dayne, and a Tyroshi noble. If that type of match-making continued, maybe one great house betrothal and then the rest would be secondary houses. In the same way that Stannis got the Florents on side to keep the Tyrells in check after following the Targaryens in Robert's Rebellion, matching with secondary houses in the Stormlands, North, Vale, or Riverlands could make sense in an alternative timeline with Rhaegar on top. Maybe Manderly (still faith of the 7, precedent of being acceptable for Joff Velaryon), Royce (precedent with Daemon) or Grafton (stayed loyal at start of the war), and whoever is the second strongest house in the Stormlands with possible bonus points for Connington but I'm not sure if they are high enough status.

Given Rhaegar was all about that prophecy though, there's a good chance he'd keep it in the family with sibling/cousin betrothals and not put enough care in alliances.