Is a fox in competition with a wolf?

No. Foxes are an entirely different species from wolves. Foxes belong to the Tribe Vulpini whereas the wolf belongs to the Tribe Canini. The wolf and the fox share the same family of Canidae, but that's as far up the family tree that they are related.

Foxes do not form packs like wolves, thus would rather work or hunt independently or with another partner. Foxes are often half the size that wolves are. Foxes hunt much smaller prey as well, and can only scavange carcasses of game that are larger than themselves that other large carnivores have brought down.