Given the size of their ships, my guess is that the
First Federation are kind of an older power that for the most part is focusing on its own separate development apart from some of the younger powers. They're not Metrons or anything like that, and they'll trade with others, but they can take care of themselves and prefer to keep apart. I think the Dominion would certainly pose a threat to them, but they just never thought things were quite dire enough to turn back.
The
Gorn are either now part of the Federation (Cestus III I think is a Federation world), were conquered by the Klingons (that'd be pretty cool so show them as tough, and their empire a multi-special one), or went
extinct.
The
Sheliak are too far away to care and corporate negotiations were taking too long. They, like their fellow non-humanoid Tholians, decided to wait it out. Maybe they'd attack later when all sides were weakened.
The
Tzenkethi and the
Jarada were waiting to see which way the wind was blowing before committing. They feared the Dominion but were also being wooed by it. The Tzenkethi especially liked the Dominion's genetic-engineering techs (see DS9 Relaunch novels), and the Jarada the highly respectful overtures of the Vorta.
The
Nausicans are more mercenaries and mobsters than soldiers. They deal with civilians, not militaries.
The
Tamarians were a little far off and communication still an issue there. They kept to themselves and their neighbors unsure how the larger game would play out. Federation diplomats got them to send aid and maintain regional order but they didn't commit to war.
The
Talarians did fight on the side of the Federation Alliance but on the Breen front, away from Cardassia.