This is contained in $\S$6 (Corollary 6.7) of Ross Street's paper

Street, Ross. Conspectus of variable categories. J. Pure Appl. Algebra 21 (1981), no. 3, 307--338.

where it is shown more generally that each 2-category of the form $\mathrm{Hom}(\mathcal{C}^\mathrm{op},\mathbf{CAT})$, where $\mathcal{C}$ is a small 2-category, has a Yoneda structure.

Note that $\mathrm{Hom}(\mathcal{C}^\mathrm{op},\mathbf{CAT})$ is the 2-category whose objects are the pseudofunctors $\mathcal{C}^\mathrm{op} \to \mathbf{CAT}$, whereas your 2-category of interest is the full sub-2-category of one of these on the strict 2-functors. But as this full sub-2-category is biequivalent to $\mathrm{Hom}(\mathcal{C}^\mathrm{op},\mathbf{CAT})$, the Yoneda structure should transport to one on the full sub-2-category.

This is contained in $\S$6 (Corollary 6.7) of Ross Street's paper

Street, Ross. Conspectus of variable categories. J. Pure Appl. Algebra 21 (1981), no. 3, 307--338, doi: 10.1016/0022-4049(81)90021-9.

where it is shown more generally that each 2-category of the form $\mathrm{Hom}(\mathcal{C}^\mathrm{op},\mathbf{CAT})$, where $\mathcal{C}$ is a small 2-category, has a Yoneda structure.

Note that $\mathrm{Hom}(\mathcal{C}^\mathrm{op},\mathbf{CAT})$ is the 2-category whose objects are the pseudofunctors $\mathcal{C}^\mathrm{op} \to \mathbf{CAT}$, whereas your 2-category of interest is the full sub-2-category of one of these on the strict 2-functors. But as this full sub-2-category is biequivalent to $\mathrm{Hom}(\mathcal{C}^\mathrm{op},\mathbf{CAT})$, the Yoneda structure should transport to one on the full sub-2-category.