The oxygen isotope (¹⁶O/¹⁸O) effect (OIE) on the pseudogap (charge-stripe ordering) temperature T* is investigated for the cuprate superconductor La_2−xSr_xCuO₄ as a function of doping x by means of x-ray absorption near edge structure studies. A strong x dependent and sign reversed OIE on T* is observed. The OIE exponent α_T* systematically decreases from α_T* = −0.6(1.3) for x = 0.15 to α_T* = −4.4(1.1) for x = 0.06, corresponding to increasing T* and decreasing superconducting transition temperature T_c. Both T*(¹⁶O) and T*(¹⁸O) exhibit a linear doping dependence with different slopes and critical end points [where T*(¹⁶O) and T*(¹⁸O) fall to zero] at x_c(¹⁶O) = 0.201(4) and x_c(¹⁸O) = 0.182(3), indicating a large positive OIE of x_c with an exponent of α_xc = 0.84(22). The remarkably large and strongly doping dependent OIE on T* signals a substantial involvement of the lattice in the formation of the pseudogap, consistent with a polaronic approach to cuprate superconductivity and the vibronic character of its ground state.