This entry in the Joyal catlab claims without proof that in a category $\bf V$ which is a "variety of algebras" the two classes $(LLP(Epi), Epi)$ form a weak factorization system. I interpret this claim in the following way:
$\bf V$ is monadic over $\bf Set$, i.e. there exists an adjunction $F\colon \mathbf{Set}\leftrightarrows \mathbf{V}\colon U$ such that $U$ is faithful and algebras for the monad $UF$ are precisely the category $\bf V$.
If I define $Epi\subset\hom(\mathbf V)$ to be the class of arrows $f$ such that $Uf$ is a surjective function in $\bf Set$, and $LLP(Epi)$ to be the left orthogonal of this class, then these classes are orthogonal and every arrow in $\bf V$ can be factored as a composition $X\xrightarrow{LLP(Epi)}A\xrightarrow{Epi}Y$.
The two classes are orthogonal almost by definition; it's rather easy to prove the characterization given for $LLP(Epi)$: codomain retracts of an inclusion $A\hookrightarrow A\coprod FX$, where $X$ is any set.
My problem is that I've no clue on how to prove the factorization property: the only thing I can think to appeal is some general, indirect result on existence, but I'm looking for something more explicit.