A very incomplete list of papers and books on groupoids (or that mention groupoids) arranged chronologically (by publication date)

1912

Brandt, H., Zur Komposition der quaternaren quadratische Formen, Journal fur reine und angawandte Mathematik, 143 (1913), 

1924

Brandt, H., Der Kompositionsbegriff bei den quaternaren quadratischer Formen, Math. Annalen, 91 (1924), 

Brandt, H., Bilineare Transformationen quaternarer quadratischer Formen, Math. Zeitschrift, 20 (1924), 

1926

Brandt, H., Uber eine Verallgemeinerung des Gruppenbegiffes, Math. Ann. 96 (1926), 360-366.

   Brandt discusses transitive groupoids and this appears to be the first paper where groupoids were introduced. He defined groupoids because he was generalising a composition of binary quadratic forms due to Gauss to quaternary quadratic forms.

1927

Loewy, A., Uber abstrakt definierte Transmutationssysteme oder Mischgruppe, J. Reine Angew. Math. 157 (1927), 239-254.

    According to A.H. Clifford and G.B. Preston in their book The Algebraic Theory of Semigroups, Loewy introduced what he called a mixed group which turned out to be equivalent to the groupoid introduced by Brandt.

Loewy, A., Neue elementare Begrundung und Erweiterung der Galoisschen Theorie, S.-B. Heidelberger Akad. Wiss. Math. Nat. Kl. 1925 (1927), Abh. 7

    In describing the relations between subfields of a field K via morphisms of K, groupoids appear in the Galois theory.  The isotropy groups of the constructed groupoid turn out to be the Galois groups.

Schmidt, F.K., Bemerkungen zum Brandtschen Gruppoid, Sitzungsberichte Heidelberg 8 (1927) 91-103.

1928

Brandt, H., Idealtheorie in einer Dedekindschen Algebra, Jahresbericht Deutsch. Math. Verein 37 (1928), 5-7.

1929

Baer, A., Zur Einfuhrung des Scharbegriffs, J. Reine Angew. Math. 160 (1929), 199-207.

    Baer formulated the structure for a ternery operation t on a group G.  The set of subsets of G closed under t is a groupoid.

Brandt, Primzahlzerlegung in einer Dedekindschen Algebra, Schweizerische Naturforschende Gesellschaft Verhandlung, 28 (1929), 288-190.

1930

Brandt, H., Zur Idealtheorie Dedekindscher Algebren, Comment. Math. Helv. 2 (1930), 13-17.

1937

Suschkewitsch, A., Theory of Generalized Groups, Gos. Nauk. Tekh. Izd. Ukranii, Karkow, 1937.

    In Russian.  According to A.H. Clifford and G.B. Preston in their book The Algebraic Theory of Semigroups, Chapter 5 of this book discusses Loewy's mixed groups and Brandt's groupoids and their equivalence.

1940

Brandt, H., Uber die Axiome des Gruppoids, Vierteljschr. Naturforsch. Ges. Zurich 85 (1940), 95-104.

1942

Clifford, A.H., Matrix representations of completely simple semigroups, American Journal of Math. 64 (1942) 327-342.

    Clifford showed that the Brandt (connected) groupoids with a zero adjoined were semigroups isomorphic to B(G,I).

1948

Fox, Ralph H., Homotopy groups and torus homotopy, Ann. of Math., 49 (1948), 471-510.

    The earliest use of the the term fundamental groupoid in the literature?

1957

Ehresmann, C., Gattungen von lokalen Strukturen, Jahresbericht der Deutschen Mathematiker-Vereinigung, 60 (1957) 49-77.

    This paper associates a groupoid with an inverse semigroup and a natural partial order.

Gray, J.W., A theory of pseudogroups with applications to contact structures, Technical Report No. 65 Applied Mathematics and Statistics Laboratory Stanford University, California, 1957.

    This report describes the relationship between a topological groupoid and a pseudogroup.

1958

Haefliger, A., Structures feuilletees et cohomologie a valeur dans un faisceau de groupoids, Comment. Math. Helv. 32 (1958) 248-329.

1960

Hasse, M., Einige Bemerkunge uber Graphen, Kategorien und Gruppoide, Math. Nachr. 22 (1960) 255-270.