In Deep Space 9, there was an ongoing plotline involving a war between Cardassia and The Klingon Empire. Obviously the did have the war, but the crux of my question is how they managed to do so without having the support of The Federation. Most maps I've seen of the Star Trek universe, such as this one, show at least a good portion of Federation space separating the Klingons and the Cardassians. Without Federation support how did the Kligons establish a supply line without also making war with the Federation?
They don't seem to share a common border. Despite the warlike posture of both races and the potential for conflict between them, the sole incident (dismissed by Garak as a "minor skirmish", albeit a bloody one that lasted 20+ years) was over a single Nebula located inside Federation space, something that seems vanishingly unlikely if they had any common contact.
In The Way of the Warrior, we see a Klingon task force making preparations to invade Cardassian space. In order to reach the Federation Cardassian border, they need to pass through Federation space.
Since the Federation and Klingons are allies (courtesy of the Khitomer Accords), presumably the Klingons can freely pass through Federation space to wherever they need to go, up to and including the Cardassian border.
SISKO : Welcome to Deep Space Nine, General. Is there something I can do for you?
MARTOK : It has been a long journey. My men require shore leave.