The planet Vulcun in Star Trek is not in our Solar System. That Vulcun was a hypothetical planet which has never been proven to exhist.

But Vulcun in Star Trek is an rocky planet orbiting Epsilon Eridani, and it would be closer to it's star then Mercury is to our Sun.
This is because the star Epsilon Eridani is an orange K type star. It's life zone is much closer to the star then ours is.

Since we have yet to begin to discover Earth sized planets, it's not out of the question that an Earth like planet could be in that star's life zone.

Live long and Prosper.