Whether or not the LNAW has an official military has never been specified, as far as I know... My understanding of the situation though is that the worlds of the LNAW have an alliance/defense pact going on. So each world would have its own military forces, which could be called together to fight alongside each other.. but are otherwise independent. That being the case.. the makeup of their forces would likely vary from world to world, as each of the worlds differs in how it treats slaves/morphs. Only stories i'm aware of that touch on the LNAW are Leanna's Story and Phoenix, and then Star Dancer: Reunions... which is very much unfinished, and barely even started. So questionable whether that should be included at all.

And eh, not too odd that we don't know too much about the LNAW. the stories stay mostly in the federation, with the exceptions in Leanna's Story and Phoenix with people from the LNAW coming into the federation... and bringing info about their specific worlds. So while the LNAW are a major(ish) faction in-universe... as far as the slice we see, they're fairly minor. Pirates and HF are both more prominent.