Over the course of 3000 years, why hasn't Jax reformed an army to fight the void by now?

It's fairly simple, actually. Not just anyone can fight the Void. Shurima had to assemble an army of demigods leading legions of ordinary soldiers to fight back the Void, and suffered heavy losses, both human and Ascended. Just LOOKING at the Void drives anyone without a strong will completely mad. I think he's less concerned about making an army to fight the Void and more-so just trying to consistently assess what the current odds of surviving it are.

Building an army to fight the Void may not have been his objective for the last 3000 years

also where do you get 3000 years? Not saying you're wrong, just curious on the hard number.

You need large populations to support large armies. Humans are also short lived

I think Jax's requirements for joining his "army" is to beat him in battle, which nobody has done up until now.

That's probably why. If they can't even beat him, how do they expect to beat the Void?

Because there wouldn’t be a story then. Or at least not the one Riot wants to tell.

I think Jax for most of that time didn't care and some story beat is what will encourage him not to give up on his former home. (Maybe a meet up with Kai'Sa?)

They could have a good mentor-student relationship I think.

But honestly though, Jax could have (and logically should have) gathered an army, however he doesn't need one, all they need is Jax himself, Zeliean (the other Icathian that wants to stop the Void), Kassadin (Wants Malzahar Dead), Kai'Sa (survived inside the Void) and Rengar (Vastaya and can keep up toe to toe with Kha'Zix)