Here is an approach that has the advantage of avoiding to go in the details of a concrete construction. More explicit constructions of the abelianization might give results under different set of assumptions, probably a bit more general, but would require more work. I don't think there really is a "minimal set of assumptions" as there might be different constructions that works under different set of assumptions. for e.g. here I'm putting light assumptions on $\otimes$ and strong assumptions on $C$, but stronger assumption on $\otimes$ (like preserving all colimits in each variables) might give rise to simpler explicit construction that require less assumption on $C$ (e.g. I would guess only the existence of countable colimits).
Theorem: Let $C$ be a symmetric monoidal locally presentable category in which the tensor product is accessible. Then the categories of (commutative) monoid in $C$ is locally presentable.
Proof: The categories of monoids or commutative monoids can be expressed as Cat-enriched limits of powers of $C$ and map induced by the tensor product between them. So as long as the tensor product is accessible, these are accessible categories because Cat-enriched limits of accessible categories and accessible functors between are accessible. As these category have all limits (they are created by the forgetful functor to $C$) they are locally presentable categories.
Corollary: Let $C$ be a locally presentable symmetric monoidal category in which the tensor product is accessible. Then the abelianization functor exists.
Proof: This follows from the special adjoint functor theorem applied to the forget full functor from commutative monoids to moinods. Fix $\kappa$ such that $\otimes$ is $\kappa$-accessible. The forgetfull functor commutes to all limits and $\kappa$-directed colimits, i.e. it is an accessible right adjoint , so it has a left adjoint by the special adjoint functor theorem.