I assume that with ``monoidal ∞-groupoid'' you mean an $E_1$-space. In this case the answer is yes. It is well known that $E_1$-spaces can be modeled by functors

$$X:\Delta^{\mathrm{op}}\to \operatorname{Space}$$
satisfying the Segal conditions. Now if you are given an $\infty$-category $\mathcal{C}$ you can define a simplicial space
$$s(\mathcal{C}): [n]\mapsto\operatorname{Map}_{\operatorname{Cat}_∞}(\Delta^n,\mathcal{C})\,.$$
In fact this functor is fully faithful and identifies $\operatorname{Cat}_∞$ with the category of complete Segal spaces. For the following we won't need all this though - we will use only that it takes values in Segal spaces (which follows immediately from $\Delta^n\amalg_{\Delta^0} \Delta^m\simeq\Delta^{n+m-1}$ in $\operatorname{Cat}_∞$).

Now let $x\in\mathcal{C}$ be an object of $\mathcal{C}$. Then we can define the simplicial space
$$ \operatorname{End}_{\mathcal{C}}(x):\Delta^{\mathrm{op}}\to \operatorname{Space}\qquad [n]\mapsto \{x\}\times_{\operatorname{Map}(\{0,\dots,n\},\mathcal{C})} \operatorname{Map}_{\operatorname{Cat}_∞}(\Delta^n,\mathcal{C})\,.$$
That is it sends $[n]$ to the (∞-)groupoid of functors $F:\Delta^n\to \mathcal{C}$ that sends all objects to $x$. It is easy now to see that $\operatorname{End}_{\mathcal{C}}(x)$ satisfies the Segal conditions and so it is an $E_1$-space.

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This takes care of your preliminary question. To go back to your main question, the functor $(\mathcal{C},x)\mapsto \operatorname{End}_{\mathcal{C}}(x)$ obviously preserves all limits and filtered colimits, and so it has a left adjoint $B$ exactly as you wanted. To get a more ``concrete'' description $B$ sends an $E_1$-space $X$ to the ∞-category corresponding to the completion of $X$ seen as a Segal space. That is
$$BX:=\int^{[n]\in\Delta^{\mathrm{op}}} X([n])\times \Delta^n$$
where the coend is computed in $\operatorname{Cat}_∞$.

With more care one can show that $B:E_1-\operatorname{Space}\to(\operatorname{Cat}_∞)_{\Delta^0/}$ is fully faithful with essential images those arrows $\Delta^0\to\mathcal{C}$ that are essentially surjective (that is such that $\mathcal{C}$ has only one equivalence class of objects). Indeed this is a special case of the equivalence between the ∞-category of Segal spaces and the ∞-category of flagged ∞-categories.