This book gives a basic summary of the conflict.
Demon The fallen, Storyteller’s Companion page 68-69
Demon and the kindred are competitors for the same prize, mankind and
all it has to offer. Both groups are highly territorial, prideful and
more than willing to resort to violence if threatened.
So far, any attempts at détente between a kindred and a court in a
given town or city have broken down into outright conflict. This is
not to say that all demon-vampire relations are ultimately
destructive. Some of the fallen, particularly of the Reconciler
faction, have forged ties with individual vampires, hoping to use
their lore to restore the kindred’s humanity in exchange for knowledge
and support as the demon try to further their agenda worldwide.So far,
these efforts at returning the undead to life have failed.
As such, when they enter vampire territory have a senior member of the vampire’s domain explain things to them. The sheriff or prince, probably. They can say this is their territory, their region, and the PCs are not allowed to hunt there. That said, they respect the value of diplomacy between supernatural nations, and are supportive of you killing the earthbound who is causing them problems and stealing their vampires and riling up the locals. They can offer you a neophyte with local knowledge to support them in staying on the right side of polite society and destroying the earthbound.
You can decide their personality. They could offer them the Earthbound’s territory if they feel it is tainted, or offer other perks for resolving the issue, or they could be condescending and rude, but regardless, vampires and demons don’t need to fight. Vampires are very manipulative, and they are likely fine with demons fighting other demons rather than expending their own vampires to fight the earthbound. The cause of their conflict is fighting over a shared resource. So long as the demons don’t start gathering faith from mortals they can cooperate.