The question (under the relaxed notion of operad in the original post) will be settled negatively if we show that the category $\mathrm{Grpd}$ (internal groupoids in $\mathrm{Set}$) isn't monadic over $\mathrm{Set}$ for any choice of underlying functor $U: \mathrm{Grpd} \to \mathrm{Set}$. (Note: I am using $\mathrm{Grpd}$ to denote the category, not the putative operad of the OP.)
Categories that are monadic over $\mathrm{Set}$ enjoy strong exactness conditions: they are Barr-exact. In particular, a category that is monadic over $\mathrm{Set}$ in some way is a regular category, and I claim that $\mathrm{Grpd}$ is not regular. In fact I will give an example of a regular epi in $\mathrm{Grpd}$ that is not stable under pullback.
Let's take a connected groupoid $G$ with two objects $0, 1$ whose automorphism group at either object is say $\mathbb{Z}$ (it won't matter much what the automorphism group is, but to be definite let's take this one). Let $A$ be the coproduct of two copies of $G$, and let $B$ be the quotient of $A$ obtained by identifying $1$ of the first copy with $0$ of the second. Then $B$ is a connected groupoid with three objects $0, 1, 2$ and with automorphism group the free group $\mathbb{Z} \ast \mathbb{Z}$, with the object $1$ being the identification of the two aforementioned objects of $A$.
Let $C$ be the full subgroupoid of $B$ on the objects $0, 2$, with inclusion $i: C \to B$. Consider what you get by pulling back the quotient map $p$ along $i$:
$$\begin{array}{ccc}
P &\to & A\\
q \downarrow & & \downarrow p \\
C & \stackrel{i}{\to} & B
\end{array}$$
The domain $P$ sits inside $A$ as the full subcategory whose objects are $0$ of the first copy of $G$ and $1$ of the second copy. There is no way $q: P \to C$ can be a quotient map, since the images of the two automorphism groups of $P$ under $q$ are disjoint (hence miss the arrows connecting $0$ to $2$ in $C$.
Edit: Nor is $\mathrm{Grpd}$ described by a multisorted Lawvere theory. For example, the category of algebras of a $2$-sorted Lawvere theory is monadic over $\mathrm{Set}/2$, and such categories are again regular. One result along these lines is that if $\mathbf{D}$ is finitely complete and finitely cocomplete and coequalizers in $\mathbf{D}$ split, and $U: \mathbf{C} \to \mathbf{D}$ is monadic, then $\mathbf{C}$ is regular; see for example Proposition 4.6 here. This applies in particular to any $\mathbf{D} = \mathrm{Set}/X$.
