Optical vortices in planar geometries are a universal wave phenomenon, where electromagnetic waves possess topologically protected integer values of orbital angular momentum (OAM). The conservation of OAM governs their dynamics, including their rules of creation and annihilation. However, such dynamics remained so far beyond experimental reach. Here, we present a first observation of creation and annihilation of optical vortex pairs. The vortices conserve their combined OAM during pair creation/annihilation events and determine the field profile throughout their motion between these events. We utilize free electrons in an ultrafast transmission electron microscope to probe the vortices, which appear in the form of phonon polaritons in the 2D material hexagonal boron nitride. These results provide the first observation of optical vortices in any 2D material, which were predicted but never observed. Our findings promote future investigation of vortices in 2D materials and their use for chiral plasmonics, toward the control of selection rules in light-matter interactions and the creation of optical simulators of phase transitions in condensed matter physics.