What involvement did Jonathan Edwards have with the Conspiracy of 1741?

One claim is wrong. Evidence for a second claim is over-all circumstantial.

From the OP:

The article proceeds to draw inferences from the text of the speech, descriptions of the circumstances, and a connection to "prosecuting attorney William Smith" to assert that the mania in New York that summer was the subject of his oration. Finally the article concludes:

Ultimately absolved by their minister, the jubilant people in Enfield were free; but thrilling sermons in Connecticut could be no solace to the tortured in New York.

A simple peak at the time line disproves this. The sermon was written about a month before the infamous trial took place. Some of the accused were burned at the stake, thus "tortured."

The second claim:

This topical sermon [Sinners in the Hands of an Angry God] is a bitter jeremiad against the "New York Negro rebels" who were then being executed for plotting to burn the village of New York to the ground.

The sermon was written after the start of the Conspiracy, which started in the Spring of 1741. It isn't known if Edwards knew about it, but it can be assumed that he did, since it caused wide-spread hysteria.

As far as direct evidence linking the event to the penning of the sermon, it appears that the evidence thus far available is only circumstantial.

Edwards was a slave owner and his views and sermons about slavery waxed and waned over his career, but he supported an end to the international slave trade and the humane treatment of slaves generally.

Not much has been written about Edwards, which is why it is difficult to know if there might be a relationship. In 1995, a great deal of new information was discovered about him in the form of personal letters. Kenneth P. Minkema has been studying these letters and Edwards other personal writings and published a paper in 2009, Jonathan Edwards's Defense of Slavery for the The Massachusetts Historical Review Vol. 4, Issue NA. I am referencing his work unless otherwise stated.

Edwards wrote sermons in the 1730s clearly referencing slavery themes and then seemed to abandon the theme until the summer of 1741. It is possible he is talking about slavery in Sinners in the Hands of an Angry God, but it is difficult to prove. (Minkema would suggest it is most likely not about slavery and Edwards pro and anti slavery views were influenced by personal considerations, mainly a personal attack on another preacher who was unpopular with his followers, as well as religious and ethical concerns.)

What can be proven is that several weeks after writing the famous sermon, he wrote down thoughts strongly condemning slavery. His sermons written directly after Sinners in the Hands of an Angry God are thus absolutely anti-slavery.

These anti-slavery sentiments coincide with the Conspiracy of 1741 and an abolitionist movement within his church community. The timing would suggest the event did influence Edwards to write other important sermons and influenced the Great Awakening. The event may not have impacted this particular sermon, though.

I, too, was looking for confirmation of the article linking the Rebellion of 1741 to "Sinners". About the statement that the sermon took place before the trials and executions, I have found evidence to the contrary. Hordsman, Daniel. ”A Journal of the Proceedings in the Detection of the Conspiracy formed by some White People in conjunction with Negro and other Slaves, for burning the City of New-York in America, and murdering the Inhabitants.” Published in New York by James Parker. Howes H652, Sabin 33058. Horsmanden presided over the trial and later served on New York’s Supreme Court. Fullest account of the so-called Negro Plot of 1741, based on depositions. Reprinted in 1971 by Beacon (see GLC 4205.02)

Records show many were arranged and punished before July of that year. It is quite possible that there is a connection. However, without a primary source linking this event to the writing of "Sinners" it can still only be considered speculative.