The Easter Bunny is widely believed to have its origins in German Lutheran story telling about the Easter Hare.
Unsurprisingly, therefore, the first English language telling of this seasonal visitor that I could find using the ID method on Google is an 1856 translation of German Stories. See Red Beard's Stories for Children: Translated from the German by Cousin Fannie. Boston. Phillips, Sampson and Co page. 7):
Goes to show - Google Library has not yet scanned the earliest sources of everything.
Surprisingly, on Google Books, the term "Easter Rabbit" does not get us back any earlier than "Bunny," But then I tried an alternative spelling, and with it I got back 10 years further down the rabbit hole with Google than Bonnie with the term "Easter Bunnie": I found him hiding inside a magazine from 1881.
'Cries from all sides resounded of the "Easter Bunnie! oh, let us try to catch him, and see if he has brought us any Easter eggs,"'
Source: Godey's Magazine (1881 - p. 471) . Is 1881 the etymological origin of the English language term Easter Bunnie? Bonnie could not get back further than 1881 when she went digging on that term. So, unless an older bunny - or bunnie - hops or otherwise lickerty-spits our way, that is currently the case. Godey's is a USA publication. being a 19th century "lady's magazine" published in Philadelphia.