To narrow down your search:
- The lower bound is (probably) 1941. Dieudonne in his book "A history of Algebraic and Differential Topology" writes when discussing Gysin's exact sequence (which appeared in Gysin's 1941 paper):
"Exact sequences were not yet used in 1941 but Gysin's main result can be expressed as exactness of the following homology sequence..."
Assuming that Dieudonne can be trusted on this matter, 1941 provides a lower bound.
- The upper bound is 1945, the foundational paper
S. Eilenberg, N. Steenrod, Axiomatic approach to homology theory. Proceedings of the National Academy of Sciences of the United States of America. 31 (1945) 117–120.
were they define the long exact sequence of pairs as one of their axioms. They do not yet use the terminology "exact" and simply call it "a sequence" and define its key property which we now call "exactness." They do not refer to anybody else in the paper, so this might have been the earliest use of exact sequences as we know them. The terminology "exact sequence" is used by
Eilenberg and Steenrod in their book "Foundations of Algebraic Topology", 1952. I do not know if this was the earliest use, but at least this provides an upper bound.
One should check, of course, earlier (than 1945) papers, especially ones by by Eilenberg. I checked only one, Eilenberg's 1944 Annals paper on singular homology theory. There are no exact sequences anywhere in the paper.