I found this table (link) that mentions the manuscripts used for the book Select Letters of Saint Jerome (1933) PDF here which includes the aforementioned letter.

I see letter 77 in the Loeb, but not 78.

Here's how I went about answering your question:

The entry for Jerome's Epistulae is in the Clavis Patrum Latinorum (CPL), a guidebook to critical editions of texts from Tertullian to Bede, as number 0620. The CPL says that the most recent edition of the Epistulae was published in the Corpus Scriptorum Ecclesiasticorum Latinorum (CSEL) series as volume 88 by J. Divjak in 1981, a volume on Augustine's Epistulae nuper in lucem prolatae. They have also been published in CSEL vols 54-56, which are digitized here. These are older (1910-1918), so you should double check the 1981 volume if you can.

On p. 13 [EDIT: p. 49, reading is hard, see /u/AtomSlap 's response below] of CSEL vol 55, you see a list of the manuscripts used for the edition. The four oldest are dated to the 9th century: Colmar Bibliothèque municipale MS 33 (olim 41), British Library MS Cotton Vitellius C, Bayerische Staatsbibliothek MS lat 14500, and Vatican Library MS lat 355 and 356. This is a pretty typical date range for the earliest MSS of most texts.

Colmar MS 33 is digitized.

However the table is very confusing though according to another source it's supposed to correlate manuscripts with letters the manuscript was used for, maybe if someone could clarify what the letters next to the dates represent it would be helpful.

When you create a critical edition of a text, you need an apparatus that tells you how each manuscript differs from the text on the page. You don't want to write out the MS name every time, so you just assign each MS a letter. In the table you gave via the Imgur link, that is saying that the 6th century Laon (Lugdunensis) manuscript will be abbreviated in the critical apparatus as "Γ".

God, I feel so stupid, sorry, guys I meant #77.