No. Analyticity and the associated dispersion relations have nothing to do with microcausality.
Analyticity of the S-matrix is basic even to nonrelativistic quantum mechanics. This is thoroughly discussed in the book ''Scattering theory of waves and particles" by R.G. Newton. Only ordinary causality (i.e., causes are always prior to effects) is involved in the derivation. In particular, the dispersion relations are derived in Sections 10.3.4; see also the references on p.296. The derivation applies to any Hamiltonian dynamical description, and in particular to the Hamiltonian formulation of QFT discussed by Weinberg.
What is specific to relativistic quantum field theory is only the crossing symmetry, whose derivation depends on microcausality in the form of spacelike commutativity. See Weinberg I, Figure 6.4 (p.269) for the simplest case, and p.467 and p.554 for other occurrences.
The book by Eden et al. tries to argue that analyticity and crossing symmetry together might be enough to fix the S-matrix. A modern, more powerful endeavor with a similar goal is causal perturbation theory. There, in order to get stronger results, stronger analyticity ($S(g)$ analytic in $g$ in the linked article) and crossing assumptions (condition (C) in the linked article) are made (but analyticity is then weakened again by working only on the formal power series level). Dispersion relations are discussed in the causal framework by Scharf in his book of finite QED (in Section 2.8, starting just before (2.8.37)).