LATIN DICTIONARYLexicon Latīnum
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“agency” in Latin

The English word agency itself goes back to Latin agō, through agentia.

Riddle & Arnold A Copious and Critical English–Latin Lexicon, 1864

agency
operation: effectus (both the power of working possessed by anything and the effect): vis (power). (The words are found in this connection and order.) vis et effectus: efficientia (working). Sometimes impulsus (impulsion), appulsus (approximation of an effectual means or cause; especially of the sun: then, generally, the working of one thing on another).
Office of agent, procuratio (general term): coguitura (office of a procurator or agent, especially infiscal matters).
agency
procuratio (general term) cognitura (in fiscal matters).

Quick answer

What is the Latin word for agency?
The usual Latin words for “agency” are efficiō, volō, prōcūrātiō; also impellō, appellō, cognōscō. The right choice depends on the sense; see the distinctions below.

Entry from J. E. Riddle & T. K. Arnold, A Copious and Critical English–Latin Lexicon (rev. C. Anthon, New York: Harper, 1864), digitised by Fergus J. P. Walsh, CC BY-SA 4.0; Latin markup adapted by latindictionary.co.