A model of three-disk micromachine swimming in a quasi-two-dimensional supported membrane is proposed. We calculate the average swimming velocity as a function of the disk size and the arm length. Due to the presence of the hydrodynamic screening length in the quasi-two-dimensional fluid, the geometric factor appearing in the average velocity exhibits three different asymptotic behaviors depending on the microswimmer size and the hydrodynamic screening length. This is in sharp contrast with a microswimmer in a three-dimensional bulk fluid that shows only a single scaling behavior. We also find that the maximum velocity is obtained when the disks are equal-sized, whereas it is minimized when the average arm lengths are identical. The intrinsic drag of the disks on the substrate does not alter the scaling behaviors of the geometric factor.