According to the Oxford English Dictionary, a small printing error for the standard verb expedite gave birth to the verb expediate in 1605. Isolated typographic errors are usually ignored; but this one was recorded in a 1623 dictionary and then copied and preserved in subsequent dictionaries. It has appeared from time to time in writing ever since.
It is unlikely that the word would have persisted, even sporadically, in spoken and written English simply because it could be found in the occasional dictionary. It is more likely that different speakers and writers have, over the years, reached for expedite . . . and missed. Perhaps influenced by a host of verbs with similar endings, like appreciate and mediate, and by the stress pattern of the unimpeachable terms expedience and expedient, they have created and recreated the related word expediate.
What is remarkable is that unlike other nonstandard words (for example, irregardless, anyways, nowheres), which are primarily seen and heard in informal, less educated contexts, expediate can be found in the edited writing of reputable sources: . . .expediate the real valuation of the obstruse concept (Sir Edmund Hillary, From the Ocean to the Sky, 1980); The patients who need an expediated process . . . are the ones with high risks (Thach Nguyen, Dayi Hu, Moo-Hyun Kim, Management of Complex Cardiovascular Problems, 2007); The Defense Materials System is being operated under authority of the Defense Production Act and serves to expediate defense, space, and atomic energy programs (U.S. Senate, Hearings on Legislative Branch Appropriations for 1971).
Despite such examples, expediate has so far failed to become a standard variant and its use in circumstances that call for standard forms of English may still provoke criticism. With few exceptions, the online dictionaries that do show it describe it unequivocally as “nonstandard,” “not in use,” “obsolete,” “spurious,” or, more emphatically, “a common misconstruction of,” “an error for,” or “a worthless word for” expedite.