Let $\mathbb{T}$ be an essentially algebraic theory in which the operations are allowed to have infinite arity. It is known that not every such theory has a free/initial model; for example, there is no free complete Boolean algebra on countably many generators. However, there are infinitary essentially algebraic theories that do have initial models, for example the theory of sheaves on a small site.
My question is: is it known whether there are any conditions on an infinitary essentially algebraic theory $\mathbb{T}$ that will guarantee that it has a free/initial model? For example, will $\mathbb{T}$ have an initial model if the arities of all the operations are bounded by some cardinal?
As @varkor pointed out, if $\mathbb{T}$ is a $\lambda$-ary essentially algebraic theory for some regular cardinal $\lambda$, then $\mathbb{T}$ does have an initial model. This follows (e.g.) by (the proof of) Theorem 3.36 in the book Locally Presentable and Accessible Categories by Adamek and Rosicky, in which it is shown that if $\mathbb{T}$ is a $\lambda$-ary essentially algebraic theory over a $\lambda$-ary signature $\Sigma$, then its category of (set-based) models is a reflective subcategory of the category of partial $\Sigma$-structures. An easy computation then shows that the reflection of the initial/empty partial $\Sigma$-structure is the initial model of $\mathbb{T}$.