In this paper, we present a low-loss optical assembly utilizing a 90°-bent graded-index (GI) core polymer optical waveguide on vertical cavity surface emitting laser (VCSEL) based optical transceivers. The proposed assembly can replace conventional components such as mirrors and lenses for realizing subminiature optical engines applicable to on-board integration. To minimize the total insertion loss of the waveguide when connected to a high-speed VCSEL and a GI-core multimode fiber (MMF) at each end, the characteristics of the beam emitted from VCSELs are measured and taken into consideration for the waveguide design. In order to confirm the effect of insertion loss reduction by the waveguide numerical aperture control, 90°-bent GI-core polymer waveguides are fabricated applying the Mosquito method. The fabricated waveguide exhibits a total insertion loss as low as about 2 dB at 850-nm wavelength, which includes the coupling losses at both ends, bending, and propagation losses. We also investigate a way to reduce the insertion loss when a gap exists between the waveguide and VCSEL chip. We theoretically and experimentally confirm that filling the gap with a high index resin can reduce the coupling loss by 5 dB.