We study the equations of Wheeler-Feynman electrodynamics which is an
action-at-a-distance theory about world-lines of charges that interact through their
corresponding advanced and retarded Li
enard-Wiechert field terms. The equations are
non-linear, neutral, and involve time-like advanced as well as retarded arguments of
unbounded delay. Using a reformulation in terms of Maxwell-Lorentz electrodynamics without
self-interaction, which we have introduced in a preceding work, we are able to establish
the existence of conditional solutions. These are solutions that solve the Wheeler-Feynman
equations on any finite time interval with prescribed continuations outside of this
interval. As a byproduct we also prove existence and uniqueness of solutions to the Synge
equations on the time half-line for a given history of charge trajectories.