The design of artificial microswimmers is often inspired by the strategies of natural microorganisms. Many of these creatures exploit the fact that elasticity breaks the timereversal symmetry of motion at low Reynolds numbers, but this principle has been notably absent from model systems of active, self-propelled microswimmers. Here we introduce a class of microswimmer that spontaneously self-assembles and swims without using external forces, driven instead by surface phase transitions induced by temperature variations. The swimmers are made from alkane droplets dispersed in aqueous surfactant solution, which start to self-propel upon cooling, pushed by rapidly growing thin elastic tails. When heated, the same droplets recharge by retracting their tails, swimming for up to tens of minutes in each cycle. Thermal oscillations of approximately 5°C induce the swimmers to harness heat from the environment and recharge multiple times. We develop a detailed elastohydrodynamic model of these processes and highlight the molecular mechanisms involved. The system offers a convenient platform for examining symmetry breaking in the motion of swimmers exploiting flagellar elasticity. The mild conditions and biocompatible media render these microswimmers potential probes for studying biological propulsion and interactions between artificial and biological swimmers.