Quantized vortices are topological defects found in different two-dimensional geometries, from liquid crystals to ferromagnets, famously involved in spontaneous symmetry breaking and phase transitions. Their optical counterparts appear in planar geometries as a universal wave phenomenon, possessing topologically protected orbital angular momentum (OAM). So far, the spatio-temporal dynamics of optical vortices, including vortex-pair creation and annihilation, was observed only in Bose-Einstein condensates. Here we observe optical vortices in 2D materials and measure their dynamics, including events of pair-creation and annihilation. The vortices conserve their combined OAM during pair creation/annihilation events and determine the surrounding field profile throughout their motion between these events. The vortices are made of phonon polaritons in hexagonal boron nitride, which we directly probe using free electrons in an ultrafast transmission electron microscope. Our findings promote future investigations of vortex phenomena in 2D material platforms, toward their use for chiral plasmonics, quantum simulators, and control over selection rules in light-matter interactions.