Would it be theoretically possible for a planet to follow a perfectly circular orbit

No, because planets don't have any propulsion system to keep them from loosing distance from what they are orbiting.
To achieve balance in that orbit without a propulsion system, the planet must almost slip away from gravity when it's at a major radius distance, but still not fall into a collision route when at a minor radius distance. Meaning it has to be an ellipse.

Mr. Immortel is almost right, the others are full of it.
If a hypothetical planet is alone in a star system and going around the star in a circle, with no other bodies of sufficient size to disturb its orbit, a perfect circle could last a very long time.

I don't see why not... the rings of saturn are almost a perfect circle, notwithstanding the pull of the sun and other planets. But if these effects are not in place, a moon formed by them could possibly have a perfectly circular orbit.

No, because an orbit has to have some eccentricity to keep from decaying. When its at perihelion, it speeds up, and this gives it the momentum it needs to make it around the sun.

I don't think it could be as perfect as the edge of a record spinning on a turntable.