In this study, we report a microfluidic approach for the measurement of mechanical properties of spherical microgel beads. This technique is analogous to tapered micropipette aspiration, while harnessing the benefits of microfluidics. We fabricate alginate-based microbeads and determine their mechanical properties using the microfluidic tapered aspirators. Individual microgel beads are aspirated and trapped in tapered channels, the deformed equilibrium shape is measured, and a stress balance is used to determine the Young's modulus. We investigate the effect of surface coating, taper angle, and bead diameter and find that the measured modulus is largely insensitive to these parameters. We show that the bead modulus increases with alginate concentration and follows a trend similar to that of the modulus measured using standard uniaxial compression. The critical pressure to squeeze out the beads from the tapered aspirators was found to depend on both the modulus and the bead diameter. Finally, we demonstrate how temporal changes in bead moduli due to enzymatic degradation of the hydrogel could be quantitatively determined. The results from this study highlight that the microfluidic tapered aspirators are a useful tool to measure hydrogel bead mechanics and have the potential to characterize dynamic changes in mechanical properties.