This is pretty standard material and I don't know of a specific reference that answers your question and only that. You will find lots of references if you search for "distributions on manifolds". Perhaps the emphasis will be on scalar distributions, not vector valued ones, but the differences are minor, once you make sure all the partitions of unity used in the theory are adapted both to an atlas of coordinates and to a trivialization of the vector bundle.

A classic reference is the book of de Rham (he only deals with scalars, forms and polyvectors):

*de Rham, Georges*, Differentiable manifolds. Forms, currents, harmonic forms. Transl. from the (1955) French edition by F. R. Smith. Introduction to the English ed. by S. S. Chern, Grundlehren der Mathematischen Wissenschaften, 266. Berlin etc.: Springer-Verlag. X, 167 p. (1984). ZBL0534.58003.

Nowadays, books mostly include a single chapter or section on the theory in the generality that they need. To see specifically distributional sections of vector bundles, see for instance:

§1.1 in *Bär, Christian; Ginoux, Nicolas; Pfäffle, Frank*, **Wave equations on Lorentzian manifolds and quantization.**, ESI Lectures in Mathematics and Physics. Zürich: European Mathematical Society Publishing House (ISBN 978-3-03719-037-1/pbk). viii, 194 p. (2007). ZBL1118.58016.

§1.1 in *Tarkhanov, N. N.*, Complexes of differential operators. Rev. a. upd. transl. from the Russian by P. M. Gauthier, Mathematics and its Applications (Dordrecht). 340. Dordrecht: Kluwer Academic Publishers. xviii, 396 p. (1995). ZBL0852.58076.

**Update:** The question about a reference for a Schwartz Kernel Theorem on manifolds is more delicate. In my understanding, it does follow from the general results on tensor products of nuclear locally convex vector spaces originally proven by Grothendieck. But certainly, in order not to get into the weeds of functional analysis every time you'd like to refer to that result, it's nice to have a handy reference. I've found a version of it in Thm.1.5.1 of Tarkhanov's book referenced above. Taking a hint from wikipedia, a more classical reference (though it covers only scalar test functions and distributions) may be Thm.23.9.2 of

*Dieudonné, J.*, Treatise on analysis. Volume VII. Translated by Laura Fainsilber, Pure and Applied Mathematics (New York) 10-VII. Boston, MA etc.: Academic Press, Inc. xiv, 366 p. {$} 69.95 (1988). ZBL0672.58044.