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• Presents periodic solution continuation for impulsive delay differential equations. • Method is based on boundary-value problem and Chebyshev series expansion. • Leads to computer-assisted proofs of branches of periodic orbits. • Proves a global branch of periodic solutions of pulse-harvested Hutchinson equation. We develop a rigorous numerical method for periodic solutions of impulsive delay differential equations, as well as parameterized branches of periodic solutions. We are able to compute approximate periodic solutions to high precision and with computer-assisted proof, verify that these approximate solutions are close to true solutions with explicitly computable error bounds. As an application, we prove the existence of a global branch of periodic solutions in the pulse-harvested Hutchinson equation, connecting the state at carrying capacity in the absence of harvesting to the transcritical bifurcation at the extinction steady state. [ABSTRACT FROM AUTHOR]