In this notes Linear Analysis on Manifolds by Chris Kottke at page 20 he has

Theorem $1.16$ (Schwartz kernel theorem, c.f. [Hör85] Thm. 5.2.1). Let $M$ and $N$ be a compact Riemannian manifolds with Hermitian vector bundles $E \rightarrow M$ and $F \rightarrow N$. On the product $M \times N$ let $E \otimes F \rightarrow M \times N$ and $\operatorname{HOM}(E, F)=E^* \otimes F \rightarrow M \times N$ be the vector bundles with fibers $E \otimes F_{(x, y)}=E_x \otimes F_y \quad$ and $\quad \operatorname{HOM}(E, F)_{(x, y)}=\operatorname{Hom}\left(E_x, F_y\right)=E_x^* \otimes F_x \quad$ respectively. Then for every linear operator $A: C^{\infty}(M ; E) \rightarrow C^{-\infty}(N ; F)$, there exists a unique distribution $K_A \in C^{-\infty}(M \times N ; \operatorname{HOM}(E, F))$ with the property that for every $u \in C^{\infty}(M ; E)$ and $v \in C^{\infty}\left(N ; F^*\right)$ $$ (A u, v)=\left(K_A, u \otimes v\right) $$ where $u \otimes v \in C^{\infty}\left(N \times M ; E \otimes F^*\right)$ is the section given by $(u \otimes v)(x, y)=u(x) \otimes v(y)$. Remark. It is a common abuse of notation to confuse the operator $A$ with its Schwartz kernel, and write $$ A u=\int_M A(x, y) u(y) \operatorname{dVol}_g(y) $$

Here [Hor85] is the book by L. Hormander, The analysis of linear partial differential operators vol. I: Distribution theory and fourier analysis.

But in the book the proof is for function on $R^n$. Is there any reference that proof the theorem in the case of vector bundles?