If all truncated $n$-point functions vanish for $n>2$ (i.e. we are dealing with a so-called generalized free field), microcausality for vacuum expectation values and at the operator level are equivalent. The former, on its turn, follows from Lorentz invariance alone in the case of scalar (but not necessarily free) fields, as shown by Pierre-Denis Methée, who was a student of de Rham (Sur les Distributions Invariantes dans le Groupe de Rotations de Lorentz, Commentarii Mathematici Helvetici 28 (1954) 225-269). If the field is interacting, this is no longer the case and then, in fact, microcausality does not follow from Lorentz invariance alone, even if positive definiteness of the $n$-point functions and the energy-momentum spectrum condition also hold. Unfortunately, though, I am not aware of any explicit example of a quantum field theory which is Lorentz-invariant but not microcausal, despite some claims in the literature that it is not difficult to build such an example. If I happen to find one, I will update my answer.
It is also important to point that there are non-Lorentz invariant quantum field theories that are nonetheless microcausal (e.g. some models coupled to a suitable external "ether" field). In such models, microcausality and the energy-momentum spectrum condition suffice to ensure that energy-momentum spectrum has a Lorentz-invariant shape and therefore has Lorentz-invariant dispersion laws (i.e. either of "mass-shell" of "light-cone" type), even in the absence of bona fide Lorentz invariance - this is a consequence of the Jost-Lehmann-Dyson representation of the two-point function, which does not rely on Lorentz invariance. This shows that the concepts of Lorentz invariance and microcausality in a quantum field theory are not equivalent, regardless of the lack of examples of Lorentz-invariant but acausal models.