QUESTION: Does anyone know which document of Pope Gregory XVI contains this approbation he bestowed on the writings of St. Alphonsus Liguori? Also, would such an approbation be equivalent to an imprimatur?
The Catholic Encyclopedia says:
[The] Church herself might be held to have conceded something to pure probabilism by the unprecedented honours she paid to the Saint in her Decree of 22 July, 1831, which allows confessors to follow any of St. Alphonsus's own opinions without weighing the reasons on which they were based.
Now, this helps the search, because we just need to find that decree on that date. It seems that the decree aforementioned decree is the same thjat canonized him. However...
There is no document from Gregory XVI on vatican.va on that date: https://www.vatican.va/content/gregorius-xvi/it.index.4.html
The english site for the Congregation for the Causes of Saints has no documents from before JPII and nothing earlier than 1999 otherwise: https://www.vatican.va/content/romancuria/en/congregazioni/congregazione-delle-cause-dei-santi/documenti.html
The Italian site for the same dicastery bizarrely only lists 4 canonizations from Gregory XVI, all in the month of May, not July: http://www.causesanti.va/it/celebrazioni/canonizzazioni/1839/05.gregorio-xvi.html
So, it seems english internet searches will not turn up anything.
As for the latter part of your question, would such an approbation be equivalent to an imprimatur? The answer is no. An imprimatur, meaning "let it be printed" signifies approval for publication, usually obtained by the ordinary of the place of publication, printing, or authorship. It does not imply the agreement of the one giving the imprimatur with the author or even the entire work. Nor is it a declaration the work is necessarily free from error. St. Alphonsus would have had all his works published with permission of his superiors in his own lifetime.
09-Mar-2022 I found it!
My copy of Denzinger 43rd ed. came in today. It contains the Response of the Sacred Penitentiary to the Archbishop of Besancon, dated July 5, 1831, numbered 2725-2727 (new numbering).
- May a professor of sacred theology safely hold and teach the opinions that Blessed Alphonsus Ligouri teaches in his moral theology?
- Or should a confessor who follows all the opinions of Blessed Alphonsus Ligouri in the practice of the tribunal of holy penance be disturbed for the sole reason that the Apostolic See has found nothing deserving of censure in his works?
The confessor, in regard to what is in doubt, reads the works of the Blessed Doctor only to know his doctrine, accurately, without weighing the sources or reasons that support these various opinions; but he judges that he conducts himself in a safe manner because he can reasonably judge that a doctrine that contains nothing worthy of censure would be safe and sound and in no way contrary to the holiness of the Gospel.
Response (confirmed by the supreme pontiff on July 22, 1831): To 1: Yes, yet those who follow the opinions handed down by other approved authors should not be considered blameworthy.
To 2: No, taking note of the reasoning of the Holy See regarding the writings of the servants of God in the process of canonization. 
You can also find (only in Latin) documents on St Alphonsus from the Actae Apostolicae Sedis here
Other documents mentioned by name in the introductory paragraph to this section are: Benedict XIV, De Servorum De Beatificatione and Sanctitas et Doctrina (May 26, 1839); Decree of the Sacred Congregation of Rites of May 18, 1803; decree of Pius IX Inter eos qui of March 23 1871.
In that final decree, Pius IX said:
Moreover, he clarified obscure matters and resolved those that were uncertain, while he provided, between the entangled opinions of theologians, whether more lax or more rigorous, a sure path upon which directors of the Christian Faithful could proceed unharmed
: Toke, Leslie. "Pope Gregory XVI." The Catholic Encyclopedia. Vol. 7. New York: Robert Appleton Company, 1910. 6 Mar. 2022 http://www.newadvent.org/cathen/07006a.htm.
: Hardon, John. "Imprimatur." Modern Catholic Dictionary. 6 Mar. 2022 http://www.therealpresence.org/dictionary/adict.htm>
: Denzinger, Heinrich. Compendium of Creeds, Definitions, and Declarations on Matters of Faith and Morals. 43rd Ed. San Francisco: Ignatius Press, 2012. No. 2725-2727. p 560.