It took me several tries to properly understand the question.
You may be thinking about the Lagrange points. There are a set of equilibrium points associated with every pair of bodies in simple orbits around each other where third (light) bodies may be placed. The ones you are looking for are L1 and L2.
They are however,
- Not stable, which means you would need active station-keeping. (L4 and L5 are stable, but they make point (3) below much much worse.)
- Not usable for building a space elevator in the Earth--Sun system, because the moon will crash into the cable from time to time.
- Much further away than a geo-stationary orbit.
If you do not mean the Lagrange points, then the short answer is no.
Every thing in orbit around the Earth is also in orbit round the sun. (Indeed, the gravitational force exerted by the Earth on the Moon is considerably less than that exerted by the sun.)
But the fact that an object orbits the sun does not prevent it from being acted upon gravitationally by the Earth, so it must also be in orbit around the Earth if the situation is to be stable.