Consider an infinitary algebraic theory.
It can be represented by a category $\mathcal{T}$ with finite products, where models in a category $\mathcal{X}$ are functors $\mathcal{T}\to \mathcal{X}$ preserving finite products and morphisms between models are natural transformations. By varying the category $\mathcal{X}$ we can get models of $\mathcal{T}$ in different categories.
It can also be represented by a monad $T$ on $\mathsf{Set}$ given by free-forgetful-adjunction.
How can we do the same thing with monads? I.e: how can we get a monad $T'$ on a category $\mathcal{X}$ from the monad $T$ on $\mathsf{Set}$, such that the Eilenberg-Moore category of $T'$ is equivalent to the category of models of $\mathcal{T}$ in $\mathcal{X}$?
Assuming you mean finitary algebraic theories, since you talk about finite products (though an analogous statement holds for small products), it is proven on the nLab page multisorted Lawvere theories that given a finitary $S$-sorted algebraic theory $\mathcal T$, if $\mathcal C$ is cocomplete and admits finite products distributing over colimits, then the forgetful functor $\mathrm{Mod}(\mathcal T, \mathcal C) \to \mathcal C^S$ is monadic, i.e. equivalent to the category of algebras for a monad on $\mathcal C^S$.