There are a few ways to go about this.

First, let us observe that the surjections in $\mathbf{Set}$ are precisely the maps that have the left lifting property with respect to the inclusion $\emptyset \hookrightarrow 1$. Thus, by the usual adjointness argument, the surjections in $\mathcal{V}$ are precisely the homomorphisms that have the left lifting property with respect to the induced homomorphism $F \emptyset \to F 1$. So this weak factorisation system, if it exists, must be cofibrantly generated.

If we assume $\mathcal{V}$ is an algebraic variety in the traditional sense, then $\mathcal{V}$ will also be locally finitely presentable (or at least, locally presentable). We can then run Quillen's or Garner's small object argument to get the required factorisation.

But in fact it suffices for $\mathcal{V}$ to have finite coproducts (which is automatic if you assume the axiom of choice). Then we can proceed by hand: the factorisation of a homomorphism $f : A \to B$ is $A \to A + F U B \to B$, where $A \to A + F U B$ is the coproduct insertion and $A + F U B \to B$ is defined on $A$ by $f$ and on $F U B$ by the counit. Incidentally, I think this is what Garner's small object argument produces (cf Example 3.11 in [Understanding the small object argument]).

There are a few ways to go about this.

First, let us observe that the surjections in $\mathbf{Set}$ are precisely the maps that have the left lifting property with respect to the inclusion $\emptyset \hookrightarrow 1$. Thus, by the usual adjointness argument, the surjections in $\mathcal{V}$ are precisely the homomorphisms that have the right lifting property with respect to the induced homomorphism $F \emptyset \to F 1$. So this weak factorisation system, if it exists, must be cofibrantly generated.

If we assume $\mathcal{V}$ is an algebraic variety in the traditional sense, then $\mathcal{V}$ will also be locally finitely presentable (or at least, locally presentable). We can then run Quillen's or Garner's small object argument to get the required factorisation.

But in fact it suffices for $\mathcal{V}$ to have finite coproducts (which is automatic if you assume the axiom of choice). Then we can proceed by hand: the factorisation of a homomorphism $f : A \to B$ is $A \to A + F U B \to B$, where $A \to A + F U B$ is the coproduct insertion and $A + F U B \to B$ is defined on $A$ by $f$ and on $F U B$ by the counit. Incidentally, I think this is what Garner's small object argument produces (cf Example 3.11 in [Understanding the small object argument]).