No. The Travelers do not have any ships, and to be honest, they should probably not get any until the end of the game, or maybe even not then. However, they have had their moments with different characters, and throughout the game, they share cute moments and important memories through events, story quests, hangout quests, and events. But, as I mentioned before, nothing has been confirmed, and nor will it be confirmed, due to the nature of the game. A few significant ships for each Traveler are; Aether and Ayaka, Lumine and Tartaglia.