[@SpookySquid] I'm no co-GM, but while [i]theoretically[/i] it's possible, and I emphasize theoretically XD Here's what Wookieepedia had to say regarding lightsabers used after the Great Jedi Purge:

"With the Great Jedi Purge, lightsabers were outlawed. To further hinder surviving Jedi, Palpatine had many of the sites where Jedi traditionally collected lightsaber crystals razed or quarantined, and placed strict sanctions on any crystals available on the market that could be used on such, such as Corusca gems.[4][29] Jedi were forced to refrain from drawing their lightsabers during this era, as openly displaying their traditional weapon would only draw attention.[30] Ultimately, the only individuals permitted to wield lightsabers were Palpatine's own Dark side servants, such as Darth Vader or his Shadow guardsman, to name a few.[31] As the widespread outlawing of these weapons made an already rare artifacts now almost impossible to find save in the hands of surviving Jedi, Jedi hunters would often collect the weapons as proof of kill, either turning them over to the Empire or keeping them as trophies."

Now, since the Empire won the Battle of Yavin and the Alliance is all but destroyed, I think that'd make it even harder - if not impossible for someone to possess a lightsaber at the moment. Which is not to say it couldn't be found as the RP progressed. But again, I hold no power in the RP, the final decision is up to [@Galactic Hero] - that's just my two cents :)