  We obtain general weak existence and stability results for stochastic
convolution equations with jumps under mild regularity assumptions, allowing
for non-Lipschitz coefficients and singular kernels. Our approach relies on
weak convergence in $L^p$ spaces. The main tools are new a priori estimates on
Sobolev-Slobodeckij norms of the solution, as well as a novel martingale
problem that is equivalent to the original equation. This leads to generic
approximation and stability theorems in the spirit of classical martingale
problem theory. We also prove uniqueness and path regularity of solutions under
additional hypotheses. To illustrate the applicability of our results, we
consider scaling limits of nonlinear Hawkes processes and approximations of
stochastic Volterra processes by Markovian semimartingales.
