A frame field is a section of a frame bundle (of a manifold), hence a collection of vector fields that form a basis of the tangent space at each point.
Pietro Fré, §3.2.5 in: Gravity, a Geometrical Course, Volume 1: Development of the Theory and Basic Physical Applications, Spinger (2013) [doi:10.1007/978-94-007-5361-7]
(on the historical origin of Cartan moving frames in Cartan geometry)
Kirill Krasnov, §3 in: Formulations of General Relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press (2020) [doi:10.1017/9781108674652, taster:pdf]
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