Are items bought with exploration badges soul bound (or whatever they choose to call it in STO)?
also
Can you get a Turret from the weapons exploration vender?

I think most if not all are soulbound, but your BO's can use armor/weapons you get.

Also, yes, you can get turrets, and this is probably the best way to do so. Also, consoles.

Note that as far as I can see the exploration and pvp vendors mirror each other in terms of what they offer. Badge costs are different and I don't really know which is more time-efficient. Of course, if you're not good at PvP I can't recommend focusing on that.

Here is why I ask, I am trying to get the gear for my JT during head start for obvious reasons.

I can't seem to find a turret to save my life, and i need to find a non soulbound one so i can send it to the other char. Last time I checked the weapon vender in sol station only carried phaser arrays, photon torpedoes, and photon mines, no dual cannons or turrets.

The best place to find turrets on the Federation side, without using an Exploration or PvP vendor, is the Exchange. Many show up on there as they aren't the most desired weapons of choice across the board.