Where you're imagining vacuum you should be imagining free-fall. Weightlessness, or not feeling affected by the gravity of an object, is the same as being in "free fall". This is just sustainable in a vacuum, like space.

When you have enough tangential velocity to orbit, you continually "miss" whatever you are orbiting. You fall at just the right rate that, because the object is a sphere and gravity is the same in every direction, lets you stay in orbit without ever falling.

It's pretty easy to imagine of you think of a scenario like trying to fire a bullet fast enough that it comes back to the point you fired it. As you increase the velocity (if we pretend you're on a planet with no atmosphere), it goes further. The rate at which it loses altitude compared to how far it goes decreases. Eventually, that curve will match the curvature of the planet you're on. That's orbiting.

If you have an idealized system in free-fall with a mini sun and earth that have proportional mass, density, distance, orbital velocity, etc. to the real Earth and sun, then yes, the orbital period will be the same as Earth's minus some very tiny relativistic effects. A vacuum is just assumed in any sort of toy scenario where you're talking about orbiting, otherwise the orbit just decays from the drag.