They’ve been more or less successful, so one would assume the supersuits aren’t explicitly necessary to saving the world.
Of course, they may want to hold onto them anyway, especially if they happen to be invited to the next Arrow-verse crossover in which having the suits wouldn’t hurt.
So far, the Legends have been able to save the day without costumes by helping the Time Bureau collect some of the mythical beasts that are still out in the world post-Mallus.