been asked to officiate at a wedding, which group is legitimate to become 'ordained'

Illinois marriage law:

(750 ILCS 5/209) (from Ch. 40, par. 209)
Sec. 209. Solemnization and Registration.

(a) A marriage may be solemnized by a judge of a court of record, by a retired judge of a court of record, unless the retired judge was removed from office by the Judicial Inquiry Board, except that a retired judge shall not receive any compensation from the State, a county or any unit of local government in return for the solemnization of a marriage and there shall be no effect upon any pension benefits conferred by the Judges Retirement System of Illinois, by a judge of the Court of Claims, by a county clerk in counties having 2,000,000 or more inhabitants, by a public official whose powers include solemnization of marriages, or in accordance with the prescriptions of any religious denomination, Indian Nation or Tribe or Native Group, provided that when such prescriptions require an officiant, the officiant be in good standing with his or her religious denomination, Indian Nation or Tribe or Native Group. Either the person solemnizing the marriage, or, if no individual acting alone solemnized the marriage, both parties to the marriage, shall complete the marriage certificate form and forward it to the county clerk within 10 days after such marriage is solemnized. 

[snip]

    (b) The solemnization of the marriage is not invalidated: (1) by the fact that the person solemnizing the marriage was not legally qualified to solemnize it, if a reasonable person would believe the person solemnizing the marriage to be so qualified; or (2) by the fact that the marriage was inadvertently solemnized in a county in Illinois other than the county where the license was issued and filed. 
(c) Any marriage that meets the requirements of this Section shall be presumed valid. 

(Source: P.A. 98-597, eff. 6-1-14; 99-90, eff. 1-1-16.)