Let $T$ be a Lawvere theory. Then view the category of $T$-algebras, as Professor Blass wrote, as a class of universal algebras for a signature that corresponds to $T$ and is equationally definable, i.e. a variety. Then do what one does with these kinds of algebras:
To get coequalizers use quotients of algebras by what are called congruences (at least, that is what they are called in Universal Algebra). A congruence is an equivalence relation on the set that corresponds under your functor $A$ to the object 1 in $T$ (that is, the underlying set) that is preserved under all operations of the theory (arrows in $T$).
To get all coproducts , first obtain the initial object $t\mapsto \emptyset$. Then construct the left adjoint by defining the free $T$-algebra on some set $S$ to be the obvious structure on the set of all formal expressions $f(s_1,...,s_n)$ with $f$ an arrow of $T$ and $s_1,...,s_n\in S$. It is then clear what the coproduct of a set of free $T$-algebras is, and you can use congruences to get the coproduct on an arbitrary set of $T$-algebras.
