In E2, once the heros find the relic "Trueshot", can they immediately use it or do they have to wait until they win the encounter and have it awarded to them?

If it stated as a reward, than it becomes available in the following quest.

No, it cannot be immediately used. They have to wait until the end of the 2nd encounter, and then only if the Heroes win the encounter. It is given as a reward if the Heroes win.

The flavor text supplies the answer for you, which is read aloud when you reveal the unique search token. "Although not currently strung, this will clearly be a powerful weapon against the Overlord as soon as you can get it back to Aryhnn."

My group had the same question. After a short discussion, they reluctantly agreed that it is not usable until after the quest ends.

It has always bothered me from a thematics perspective. I have never played as a hero in this game, but this particular point has always made me a bit sorry for my opposition. I mean, they get in the dark room, grab the relic from its shiny spot, and they don't get to use it, only transport it? The game doesn't give an explanation as for why a relic found during a quest wouldn't be useable straight away, only that heroes must wait for the encounter reward phase to receive it.

Same goes for this other quest where Alric finds the Duskblade sword in the Dawnblade quest encounter II, but cannot get to use it until the encounter is over.

No problem at all from a game mechanism perspective, but thematically it is hard to comprehend in my experience. It is also a source of disagreement if said rule hasn't been made very clear to everyone.

The bowstring needs to be replaced, so they need to bring it back to town.

Or, the sword has dulled, same thing. It's at least been clarified in the responses from FFG. Unless it says in the rules that you can equip the thing you found, you can assume you can't.

It has always bothered me from a thematics perspective. I have never played as a hero in this game, but this particular point has always made me a bit sorry for my opposition. I mean, they get in the dark room, grab the relic from its shiny spot, and they don't get to use it, only transport it? The game doesn't give an explanation as for why a relic found during a quest wouldn't be useable straight away, only that heroes must wait for the encounter reward phase to receive it.

Obviously, the icon is on cooldown when the ability is first gained. Sheesh!