If a card gets reprinted, can you still use the old version of it after rotation happens?

Short version? No, but We won't know for certain till after rotation. It seems common sense they should, since that's what most games allow.

Long Version? Well, what makes a card legal in the first place? What the organized play doc for Destiny says is that proxies aren't allowed except under organizer approval in the approval of the event leader in cases of damaged/missing cards, FFG promos are legal if their original printing is legal for the event, and most importantly cards legal for the format of the event according to the holocron for that format. And the holocrons currently simply list what sets/products are legal.

So currently the answer is likely no, they wouldn't be legal assuming that the list of legal sets excludes that sets that printed the original cards. However once we rotate we will get new holocrons and hopefully some clarifications on the issue. However I don't think anyone had any issues with people running original print Doubt, tactical mastery or anything of the sort in Trilogy format the past year. This may change as we get functional errata with stuff like First Order Stormtrooper gaining a subtype with the reprint.