Small dg-categories are categories enriched over $\mathcal{V} = \textbf{Ch}(k)$, the category of chain complexes over a commutative ring $k$.

It is a general result that if $\mathcal{V}$ is (co)complete, then so is the category $\mathcal{V}\textbf{-Cat}$ of small $\mathcal{V}$-enriched categories.
See the text around Corollaries 7.3.6 and 7.3.7 of Christina Vasilakopoulou's thesis for a sketch of the proof and other references of this fact.

This paper by Harvey Wolff proves that $\mathcal{V}\textbf{-Cat}$ is cocomplete by first showing that $\mathcal{V}\textbf{-Graph}$ is cocomplete ($\mathcal{V}$-graphs are just $\mathcal{V}$-categories without identities or composition). The construction for coproducts is easy enough, but things get quite involved when constructing coequalizers even for just $\mathcal{V}$-graphs.

In the remainder of this answer, I'll just show how to construct coproducts of $\mathcal{V}$-categories.

Let $\mathcal{C}_i, i \in I$ be a collection of $\mathcal{V}$-categories for some indexing set $I$. Form a $\mathcal{V}$-category $\mathcal{C}$ in the following manner:

- $\text{Ob}(\mathcal{C}) := \coprod_{i \in I} \text{Ob}(\mathcal{C}_i)$
- For $x,y \in \text{Ob}(\mathcal{C})$ where $x \in \text{Ob}(\mathcal{C}_i)$ and $y \in \text{Ob}(\mathcal{C}_j)$, define $\mathcal{C}(x,y)$ to be $\mathcal{C}_i(x,y)$ if $i = j$ and $0$ otherwise, where $0$ is the initial object of $\mathcal{V}$.
- For $x,y,z \in \text{Ob}(\mathcal{C})$, we need to produce composition maps $$\mu_{x,y,z}\colon \mathcal{C}(x,y) \otimes \mathcal{C}(y,z) \to \mathcal{C}(x,z).$$ If $x,y,z$ are all from the same $\mathcal{C}_i$, then just use the composition maps from $\mathcal{C}_i$. Otherwise, at least one of $\mathcal{C}(x,y)$ and $\mathcal{y,z}$ is $0$. Since $0 \otimes X = X \otimes 0 = 0$ for any $X \in \mathcal{V}$, the domain of $\mu_{x,y,z}$ is $0$, so take $\mu_{x,y,z}$ to be the unique map $0 \to \mathcal{C}(x,z)$.

It's easy to check that composition as defined above is associative and unital.