  We consider a directed polymer model in dimension $1+1$, where the disorder
is given by the occupation field of a Poisson system of independent random
walks on $\mathbb Z$. In a suitable continuum and weak disorder limit, we show
that the family of quenched partition functions of the directed polymer
converges to the Stratonovich solution of a multiplicative stochastic heat
equation (SHE) with a Gaussian noise, whose space-time covariance is given by
the heat kernel.
  In contrast to the case with space-time white noise where the solution of the
SHE admits a Wiener-It\^{o} chaos expansion, we establish an $L^1$-convergent
chaos expansions of iterated integrals generated by Picard iterations. Using
this expansion and its discrete counterpart for the polymer partition
functions, the convergence of the terms in the expansion is proved via
functional analytic arguments and heat kernel estimates. The Poisson random
walk system is amenable to careful moment analysis, which is an important input
to our arguments.
