Armand Borel's Bourbaki Seminar 121 Groupes algébriques is from 1955, and uses "drapeau" (page 7). (It's online at archive.numdam.org.) This may not be the earliest occurrence, but there is a good reason for attention to the full flag variety in this context (the theory of Borel subgroups).
The concept traces back some way, to Ehresmann's thesis in the 1930s; Kolchin's work (the Lie-Kolchin theorem) uses the non-intrinsic language of upper triangular form. Hodge & Pedoe talks about Schubert spaces in general, which would be natural in the enumerative geometry tradition, for which full flags is just one of the cases.
Edit: A further data point is Chern's paper On the Characteristic Classes of Complex Sphere Bundles and Algebraic Varieties (1953), which relies on Ehresmann's work to some extent. The word "flag" is absent (though used by Chern discussing it in his Selected Papers).