I know that with conflicting passives the first player decides the order, but I'm not quite sure how it works in the following scenario:

I have a single standing Kingsroad (either winter or summer) in play with a King's Ground on it, so Vigilant and the Kingsroad's passive conflict and the first player decides. Easy enough.

If the Kingsroad is kneeling however, is there a conflicting passive or will Vigilant always initiate first since the Kingsroad can't be used unless it is standing?

A passive initiates regardless of whether it will anticipate succesful resolution - they're not bound by the rules governing triggered effects (e.g. FAQ entry 3.6). That means that The Kingsroad passive will initiate after you win a challenge, but will simply fail to either pay the cost (kneel all copies of the Kingsroad) or yield any effect (draw you a card/discard a card).

Functionally, that means that your first scenario (kingsroad standing) is no different from your second - two passives, initiating at the same time, would result in a conflict of resolution. The first player will decide which of the two initiates first.

Thanks Istaril.