In January, 1949, Shannon publishes the paper Communication in the Presence of Noise, Proc. IRE, Vol. 37, no. 1, pp. 10-21, available here, which establishes the Information Theory. In this paper, the sampling theorem is presented.
Any references about the history of the sampling theorem, its connection with Fourier theory, would be appreciated.
EDIT1: In C.E. Shannon, Mathematical Theory of Communication, Bell System Technical Journal, Vol. 27, pp. 379–423, 623–656, July, October, 1948, available here, the Sampling Theorem is Theorem 13.
Best Answer
As a start to a more comprehensive search, some notes on interpolation using the Dirichlet and Fejer kernels, close cousins of the sinc kernel, can be found in a eulogy for Fejer.
And, you yourself in your answer to MO-Q58325 present a link to a paper by J. de Seguier, published in 1892, that has a Dirichlet kernel interpolation formula and a series that looks suspiciously like a sinc interpolation with the bandwidth $\omega$.
Edit: In the old days, the sinc function was referred to as the cardinal interpolation function and sinc function interpolations as cardinal series. Here is an article (1927) by J. M. Whittaker (son of E. T.:) The "Fourier" Theory of the Cardinal Function in which you can find the nascent Whittaker-Shannon sampling theorem, but E. T. Whittaker published an earlier one in 1915 as discussed by H. D. Luke in The Origins of the Sampling Theorem.
(Also of interest) A Chronology of Interpolation: From Ancient Astronomy to Modern Signal and Image Processing (paper) (website)