I am pretty sure wikipedia is right about it being founded on May 1st 1776, but they don't reference this. Can someone provide a reliable website, preferably by authored by a historian, that can confirm this?
The source is actually linked in the article, although confusingly it has reference number  instead of  as marked in the paragraph where the statement is made:
The Illuminati (plural of Latin illuminatus, "enlightened") is a name given to several groups, both real and fictitious. Historically, the name refers to the Bavarian Illuminati, an Enlightenment-era secret society founded on May 1, 1776. The society's goals were to oppose superstition, obscurantism, religious influence over public life and abuses of state power. "The order of the day," they wrote in their general statutes, "is to put an end to the machinations of the purveyors of injustice, to control them without dominating them."
On May 1, 1776, the new organization was founded, under the name of the Order of the Illuminati. with a membership of five all told.
New England and the Bavarian Illuminati by Vernon L. Stauffer Ph.D., New York 1918
This book seems reputable, see for example this review:
The well-known reaction of American feeling from sympathy with the French Revolution to horror of its excesses and the panic made by the strident radicals of the new Democratic Clubs on this side of the ocean is not a new story, but it becomes new when read in the ample detail and interesting co-ordination of Dr. Stauffer's study--a work of admirable scholarship and excellent literary form.
Reviewed Work: New England and the Bavarian Illuminati by Vernon Stauffer by Francis A. Christie, The American Historical Review, Vol. 25, No. 1 (Oct., 1919), pp. 120-121