Given a map $f : S \to M^4$ from a compact closed not necessarily connected oriented surface to a compact oriented 4manifold, such that $f_*([S])$ is zero in $H_2(M)$, is there a compact oriented 3manifold $W$ with boundary $S$ and a map $F : W \to M$ that extends $f$?
Yes, there is. This follows from the definition of the (singular oriented) bordism group $\Omega_2(M)$ and the fact that the natural map $\Omega_2(M)\to H_2(M;\mathbb{Z})$ is an isomorphism. In fact the conclusion holds for much more general spaces $M$ (such as CWcomplexes). In more detail, $\Omega_2(M)$ denotes equivalence classes of continuous maps $f: S\to M$, where $S$ is a closed oriented surface (not necessarily connected), under the relation of bordism: two such maps $f_1: S_1\to M$ and $f_2: S_2\to M$ are declared bordant if there is a $3$manifold $W$ with boundary $\partial W = S_1\sqcup S_2$ and a map $F:W\to M$ extending $f_1\sqcup f_2: S_1\sqcup S_2\to M$ (everything up to diffeomorphism). This defines an abelian group with addition given by disjoint union. The zero element is represented by any $f:S\to M$ which bounds a map from a $3$manifold (as in your question). The natural map $\Omega_2(M)\to H_2(M;\mathbb{Z})$ sends $[f:S\to M]$ to $f_\ast([S])$. The easiest (but perhaps not the most elementary) way to see that this is an isomorphism is to examine the AtiyahHirzebruch spectral sequence for bordism (see Conner and Floyd's "Differentiable Periodic Maps", section 7), armed with the additional data that $\Omega_1 = \Omega_2 = 0$, i.e all oriented $1$ and $2$manifolds bound. 

