We derive the three-loop order renormalization group equations that describe the flat phase of polymerized membranes within the modified minimal subtraction scheme, following the pioneering one-loop order computation of Aronovitz and Lubensky [Phys. Rev. Lett. 60, 2634 (1988)10.1103/PhysRevLett.60.2634] and the recent two-loop order one of Coquand, Mouhanna, and Teber [Phys. Rev. E 101, 062104 (2020)2470-004510.1103/PhysRevE.101.062104]. We analyze the fixed points of these equations and compute the associated field anomalous dimension η at three-loop order. Our results display a marked proximity with those obtained using nonperturbative techniques and reexpanded in powers of ε=4-D. Moreover, the three-loop order value that we get for η at the stable fixed point, η=0.8872, in D=2, is compatible with known theoretical results and within the range of accepted numerical values.