Who first discovered the fact that matter and antimatter annihilate upon contact? I cannot find (yet) a place that says who specifically theorized the mutual annihilation of matter and antimatter upon contact.
I also don't know who first experimentally observed this phenomenon
In 1898, a non-rigorous non-technical paper by Arthur Schuster in Nature first 'whimsically' predicted antimatter, and coined the term, and predicted possible annihilation.... I'm not counting this.....
And Carl Anderson's experiment verified positrons by how they deflected near electric and magnetic fields, apparently...
 A: https://arxiv.org/abs/1809.04815 seems relevant, although it mostly discusses who was the first to experimentally observe annihilation. Abstract:
"In the early 1930s, the positron, pair production, and, at last, positron annihilation were discovered. Over the years, several scientists have been credited with the discovery of the annihilation radiation. Commonly, Thibaud and Joliot have received credit for the discovery of positron annihilation. A conversation between Werner Heisenberg and Theodor Heiting prompted me to
examine relevant publications, when these were submitted and published, and how experimental results were interpreted in the relevant articles. I argue that it was Theodor Heiting — usually not mentioned at all in relevant publications — who discovered positron annihilation, and that he should receive proper credit."
This article also discusses theoretical prediction of annihilation:
" Dirac interpreted unoccupied negative energy states as “holes”, which were short of negative energy, and therefore had positive energy. His theory ... is therefore often called “hole theory”. These “holes” could then be ﬁlled by ordinary electrons from higher shells, releasing energy as radiation — that is, positron annihilation. Dirac also mentioned the opposite process as a possibility (Dirac,1928): transforming electromagnetic radiation into an electron and a positron, that is, pair production 1. Dirac was well aware that protons were far too heavy to be in agreement with his theory (Dirac, 1930), but these were at the time the only known positively charged particles."
In particular, the article cites the following article: Dirac, P A M (1930), “On the Annihilation of Electrons and Protons,” Math. Proc. Cambridge 26 (3), 361–375. Dirac writes in that article:
"It has recently been proposed* that one should assume that nearly all the possible states of negative energy are occupied, with just one electron in each state in accordance with Pauli's exclusion principle, and that the unoccupied states or 'holes' in the negative-energy distribution should be regarded as protons. According to these ideas, when an electron of positive energy makes a transition into one of the unoccupied negative-energy states, we have an electron and proton disappearing simultaneously, their energy being emitted in the form of electromagnetic radiation."
Annihilation of electron and proton is certainly problematic, but it looks like it was Dirac who also predicted positron and wrote in Dirac, P A M (1931), “Quantised Singularities in the Electromagnetic Field,” Proc. R. Soc. Lond. A 133 (821), 60–72:
"It thus appears that we must abandon the identification of the holes with protons and must find some other interpretation for them. Following Oppen­ heimer,§ we can assume that in the world as we know it, all, and not merely nearly all, of the negative-energy states for electrons are occupied.  A hole, if there were one, would be a new kind of particle, unknown to experimental physics, having the same mass and opposite charge to an electron. We may call such a particle an anti-electron. We should not expect to find any of them in nature, on account of their rapid rate of recombination with electrons, but if they could be produced experimentally in high vacuum they would be quite stable and amenable to observation."
So, collectively, it looks like it was Dirac who theorized annihilation.
