We investigate the intrinsic linear and nonlinear ac orbital Hall (OH) conductivity in a two-dimensional system, arising from the transverse motion of electrons with finite orbital angular momentum in an applied electric field. Using the quantum kinetic approach, we show that the total OH conductivity comprises both interband and intraband contributions. However, the interband contribution dominates over the intraband in the high frequency regime. Our analysis predicts that the interband part of the linear OH conductivity is governed by the Fermi sea contribution. Meanwhile the nonlinear responses, including second harmonic and rectification effects, stem from the interplay between the Fermi sea and Fermi surface contributions. We find that the broken inversion symmetry in the system yields nonzero orbital angular momentum and consequently the orbital Hall response. In addition, the linear OH conductivity exhibits a resonant peak and a sign transition depending on the gap and Fermi energy values relative to the incident energy. Unlike the linear, the second harmonic OH conductivity shows two sign conversions as the incident energy approaches to the gap value and twice its value. These findings shed light on the modulation of field-driven orbital Hall conductivity with frequency, Fermi energy, and band gap.