Let $\mathcal{C}$ be a locally small category with all finite colimits, and let $\mathcal{A}$ be a small full subcategory. I wish to prove the following:
Proposition. There exists a full subcategory $\mathcal{B}$ of $\mathcal{C}$ satisfying these conditions:
$\mathcal{A} \subseteq \mathcal{B} \subseteq \mathcal{C}$
Finite colimits exist in $\mathcal{B}$ and are the same as in $\mathcal{C}$.
Every object of $\mathcal{B}$ is a finite colimit of objects in $\mathcal{A}$.
There is an obvious transfinite process that one can apply here. First, set $\mathcal{B}_0 = \mathcal{A}$. For each ordinal $\alpha$, let $\mathcal{B}_{\alpha + 1}$ be the full subcategory of $\mathcal{C}$ obtained by adding to $\mathcal{B}_\alpha$ any missing finite coproducts or coequalisers. For each limit ordinal $\lambda$, let $\mathcal{B}_\lambda = \bigcup_{\alpha < \lambda} \mathcal{B}_\alpha$.
Question. Why does this transfinite process converge? $\DeclareMathOperator{\ob}{ob}\DeclareMathOperator{\mor}{mor}$
It is easy to obtain an upper bound on the size of $\ob \mathcal{B}_\alpha$: by its very construction, $\left| \ob \mathcal{B}_{\alpha+1} \right| \le \sum_{n < \aleph_0} \left| \ob \mathcal{B}_\alpha \right|^n + \left| \mor \mathcal{B}_\alpha \right|^2$ and if we assume $\mathcal{B}_\alpha$ is non-empty, the inequality $\left| \ob \mathcal{B}_\alpha \right| \le \left| \mor \mathcal{B}_\alpha \right|$ then implies $\left| \ob \mathcal{B}_{\alpha+1} \right| \le \sup \left\lbrace \left| \mor \mathcal{B}_\alpha \right|^n : n < \aleph_0 \right\rbrace \le \left| \mor \mathcal{B}_\alpha \right|^{\aleph_0}$ which, while elegant, is not too useful since we can't bound $\left| \mor \mathcal{B}_{\alpha+1} \right|$ in terms of $\left| \ob \mathcal{B}_{\alpha+1} \right|$. It is conceivable that $\mathcal{C}$ and $\mathcal{A}$ are chosen in such a way that at each step $\alpha$, an object is added which vastly increases the size of $\mor \mathcal{B}_{\alpha+1}$.
Nonetheless, I'm fairly confident the proposition is true. Adámek and Rosický even say that "it is clear" [LPAC, p. 17] (albeit with extra hypotheses on $\mathcal{A}$ and $\mathcal{C}$) and offer no proof.