Ky. Rev. Stat. 402.050(1)(b) Marriage shall be solemnized only by ... Justices and judges of the Court of Justice, retired justices and judges of the Court of Justice except those removed for cause or convicted of a felony, county judges/executive, and such justices of the peace and fiscal court commissioners as the Governor or the county judge/executive authorizes
http://www.lrc.ky.gov/statutes/statute.aspx?id=36470
I don’t see sheriff deputy on that list.
Have the duputies been made JPs?
You're looking at the wrong statute. "Solemnized" is talking about the wedding ceremony; issuing the license is a separate, and prior, step.