Set is a vertex cover iff its complement is an independent set, therefore this problem is equivalent to counting independent sets.
Algebraic counting of independent sets is FPT for graphs of bounded bounded clique-width. For instance, see Courcelle's "A multivariate interlace polynomial and its computation for graphs of bounded clique-width", where they compute a generalization of independence polynomial. Adding up coefficients of independence polynomial gives the number of independent sets.
Graphs with maximum degree 3 can have unbounded clique-width.
Numerical counting of independent sets is tractable when the problem exhibits "correlation decay". Dror Weitz (STOC'06) gives a deterministic FPTAS for counting weighted independent sets on graphs of maximum degree $d$ when the weight $\lambda$ is
Regular (unweighted) independent set counting corresponds to $\lambda=1$ so his algorithm gives FPTAS for number of vertex covers on graphs of maximum degree 5.
His algorithm is based on building a self-avoiding walk tree at each vertex, and truncating this tree at depth $d$. Branching factor of self-avoiding walk trees determines the range of $\lambda$ for which small depth $d$ gives a good approximation, and formula above is derived by using maximum degree of the graph to upper bound this branching factor.