The idea of a relation between fundamental groups and permutations of the universal cover long predates Grothendieck and SGA. It appears implicitly already in Riemann's work on complex surfaces in 19th century. In Cauchy and Puiseux: Two Precursors of Riemann Papadopoulos even mentions earlier authors, especially Puiseux:
"Riemann defined these surfaces as ramified coverings of the plane (more precisely, of the Riemann sphere). The work of Puiseux on algebraic functions, interpreted from a topological point of view, contains in essence the combinatorics of such a surface, giving a description of how its sheets are permuted above a ramification point, and establishing the precise relation between this sheet permutation and the nature of the singularities of the algebraic equation. At the same time, Puiseux’s work makes the relation with group theory. At the expense of being anachronical, let us mention that the theory of Puiseux expresses the so-called monodromy homomorphism from the fundamental group of the Riemann sphere with a finite set deleted (the singular set of the algebraic equation) into the permutation group on $d$ symbols."
The key technical step, the path lifting theorem, was made by Weyl in 1913, and full development is due to Seifert and Threlfall's classic Lehrbuch der Topologie. Here is Dieudonné's description from A History of Algebraic and Differential Topology 1900-1960, p.296-7:
"It is quite likely that Poincare and his immediate successors, such as Tietze
and Dehn, were aware of that theorem, even if they did not formulate it explicitly (still less try to prove it!). It was stated and proved for surfaces by H. Weyl in 1913 . In 1928 Reidemeister published a short paper on fundamental groups and covering spaces  in which he only considered combinatorial manifolds. He deplored the fact that, except for dimension 2,
no previous treatment of these questions was available. His own treatment is
in fact limited to dimension 3, although he claims it might be generalized to
any dimension; it is written in the Poincare style, without genuine proofs, and
the path lifting theorem is not mentioned.
It was only in 1934 that, in their book , Seifert and Threlfall gave an
admirable and thorough elaboration of the relations between fundamental
groups and covering spaces based on the path lifting theorem: although
limited to locally finite simplicial complexes (of any dimension), it is essentially definitive, and can be extended to more general spaces with only minor modifications.
According to Theo Buehler's answer in Etymology of the name “deck transformation” thread on Math SE, the term "deck" is also due to Seifert and Threlfall. They speak of Deckbewegungsgruppe, literally, group of covering movements (from German decken, to cover). Reidemeister called them transitive permutations by the fundamental group. The "automorphisms of a fiber functor" language is of course specific to Grothendieck.