In the current version you will always have full ammo at the start of each encounter and all ships operate at 100% until destroyed. The only logistics resource at the moment is the generic supply crate which represents everything your fleet needs. There's also fuel, but it won't be usable until multiple systems are available. For now it's just extremely valuable loot.
Alex has made it clear in the past he wants to avoid having too much micromanagement in Starsector, so I doubt per-weapon resources will ever be a thing. However, the upcoming .6a update is focused on creating a bridge between the combat and campaign layers, and will -
among many other additions - introduce a new per-ship stat, 'Combat Readiness', which when low enough will have effects on your ship such as not having missiles loaded or weapons/engines having a chance to malfunction when used. Read about it
here. It will also introduce a fleet-wide stat called 'Logistics', which determines how effectively you can utilize the supplies available to you. That can be read about
here.
Keep in mind that if you can make a good case for a new/altered game mechanic in a post, a passing modder might take interest and implement it for you.