This paper addresses a challenging orbit control problem for in-orbit demonstration of propellant-free Rendezvous Proximity Operation and Docking (RPOD) CubeSat mission. The SNUGLITE-III mission, consisting of two identical 3U CubeSats, performs RPOD mission without the use of a propulsion system. Instead, we have developed an autonomous, closed-loop orbit control system that utilizes differential drag and lift in low Earth orbit (LEO) as control forces. Combined with a GPS-based cm-level relative navigation system, the proposed orbit control system generates the appropriate attitudes for the two satellites to control their relative distance within 1 km. Numerical simulations were conducted using a high-fidelity orbital propagator that includes dominant perturbations in LEO. The results confirm the successful execution of the rendezvous proximity operation maneuver of SNUGLITE-III using the proposed orbit control system. This novel orbit control method can be applied to LEO formation flying and rendezvous missions for CubeSats where thrusters are not equipped, or as a backup system for thruster-based control.