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 even more general results, in terms of corollary 2 below they would corresponds to more explicit description of the object $X_{ab}$, but I think corollary 2 below already cover most of the applications.
Theorem: Let $C$ be a symmetric monoidal locally presentable category in which the tensor product is accessible. Then the categories of (commutative) monoids 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 1: 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.
Exploiting this, one can do much better and drop the local presentability assumption by observing that "the universal example" is itself locally presentable:
Corollary 2: Let $C$ be a symmetric monoidal category with all colimits in which the tensor product preserves $\kappa$-directed colimits for some regular cardinal $\kappa$. Then the abelianization functor for $C$ exists.
Proof: Let $M_\kappa$ be the "free symetric monoidal category with all $\kappa$-small colimits generated by a monoid object $X \in M_\kappa$".
It exists, because all the structure I've put on $M_\kappa$ can be encoded as a $\kappa^+$ essentially algebraic theory (let says in a bicategorical sense for simplicity, though it would also work in a $1$-categorical sense).
Let $V_\kappa$ be the $\kappa$-ind completion of $M_\kappa$. Because Ind completion commutes to products, the tensor product on $M_\kappa$ induces a $\kappa$-accessible tensor product on $V_\kappa$, making $V_\kappa$ a symmetric monoidal category (if you prefer, ind completion being monoidal for the product, it sends monoids object to monoids object).
It follows that $V_\kappa$ is a locally presentable category with a $\kappa$-accessible tensor product, so one can apply the previous result and the universal monoid object $X \in M_\kappa$ admit a symmetrisation $X_{ab}$ in $V_\kappa$.
If now $C$ is a general symmetric monoidal category with all colimits and such that $\otimes$ preserves $\kappa$-directed colimits in each variables. Then given a monoid object $T \in C$ you have an essentially unique (strong) symmetric monoidal functor $F: M_\kappa \to C$ that preserves $\kappa$-small colimits and sends $X$ (as a monoid) to $T$.
It extends to a monoidal functor preserving all colimits $V_\kappa \to C$, in fact (as $V_\kappa$ is locally presentable), it is a left adjoint functor. It then follows from a general argument that the image of $X_{ab}$ by this functor is an abelianisation of $X$ (one just check the universal property in $C$ using the universal property of $X_{ab}$ in $V_\kappa$ and the adjunction).
