I'm writing a paper on the history of Feynman diagrams and am having trouble finding the first paper published by Feynman which features a calculation done with Feynman Diagrams.
Comprehensive sources on the history of renormalization are Cao-Schweber survey paper and Cao's book. On the diagrams specifically the source Kaiser's Drawing Theories Apart: The Dispersion of Feynman Diagrams in Postwar Physics, see also his interview to American Scientist. According to Kaiser, one of the first published diagrams (reproduced below) appears on p.772 of
Feynman, R. P.: 1949 The Space-Time Approach to Quantum Electrodynamics, Physical Review 76, 769-89.
Stueckelberg might have used them earlier in an unpublished manuscript, the topic is controversial, see Did Feynman develop QED based on Stueckelberg's manuscript? The diagrams were a computational tool for the procedure developed in Feynman's seminal renormalization papers published a year earlier. Below are the citations in the order of publication with direct links.
1) Feynman, R. P.: 1948, Space-time Approach to Non-relativistic Quantum Mechanics, Reviews of Modern Physics 20, 367-87.
2) Feynman, R. P.: 1948, A Relativistic Cut-off for Classical Electrodynamics, Physical Review 74, 939-46.
3) Feynman, R. P.: 1948, Relativistic Cut-off for Quantum Electrodynamics, Physical Review 74, 1430-38.
None of them had "Feynman diagrams" however, although there is a multiple path picture in the second one.
This paper is an analysis of Feynman's early work with Feynman diagrams, and should help you find the answer. Also look at Freeman Dyson's early papers, where he explains Feynman's techniques.