An interesting comment from 1929 is given in the Oxford English Dictionary:
If..‘normalcy’ is ever to become an accepted word it will presumably be because the late President Harding did not know any better.
OED gives the author as G. N. Clark, writing for the Society for Pure English.
"Normality" means the state of being normal.
"Normalcy" was used by Warren G. Harding in his 1920 election campaign called "Return to Normalcy." When pointed out that the word was a mistake, Harding said he couldn't find the word "normality" in his dictionary. Before his gaff, "normalcy" was used as a mathematics term. In the 90 years since Harding misspoke, the term "normalcy" has become widespread (in the USA, at least) either as an example of a mistake or as a valid synonym for "normality."
I suggest not using "normalcy" unless you know what you are doing, because it is still seen by many as a sign of ill-education.
1857, "mathematical condition of being at right angles," from normal + -cy. Associated since c. 1920 with U.S. president Warren G. Harding and derided as an example of his incompetent speaking style. Previously used mostly in the mathematical sense. The word preferred by purists for "a normal situation" is normality (1849).
(Source: Etymology Online)
America’s present need is not heroics, but healing; not nostrums, but normalcy; not revolution, but restoration; not agitation, but adjustment; not surgery, but serenity; not the dramatic, but the dispassionate; not experiment, but equipoise; not submergence in internationality, but sustainment in triumphant nationality.