To start off, here are some quotes from The Cambridge History of Western Music Theory:
To designate the scale-degree placement of these chords, Vogler introduced roman numeral designations. Although earlier theorists had proposed analogous notations and terminologies for identifying chordal scale degrees, Vogler was the first theorist to use roman numerals consistently.
This leads to a footnote:
Properly speaking, Vogler designated only the leading-tone chord (VII) in his Tonwissenschaft und Tonsezkunst [of 1776] (p. 82). It was only in his later Handbuch zur Harmonielehre (1802) that he applied roman numerals to all scale degrees.
The above two quotes are from David Bernstein's "Nineteenth-century harmonic theory" chapter (p. 780), but the latter quote points to Joel Lester's chapter on "Rameau and eighteenth-century harmonic theory" (p. 774):
Various writers invented analytical symbols to refer to the harmonies of a key. The Irish theorist John Trydell (c. 1715–76) proposed labeling chord roots by "harmonical figures" in 1766 (an invention disseminated via the 1771 edition of the Encyclopaedia Britannia); the German theorist and composer Abbé Vogler (1749–1814) used a few roman numerals later that decade; and Gottfried Weber (1779–1839) made roman numerals standard musical symbols in 1817.
Now, as for what that all means.
Vogler: Tonwissenschaft und Tonsezkunst (1776)
Notice how the above shows Vogler only using a single roman numeral, "VII," of two keys: C and A minor ("weich").
Vogler: Handbuch zur Harmonielehre (1802)
But a quarter century later, Vogler is using seven Roman numerals. Note especially the following excerpt in the middle paragraph:
Dann wird man eingestehen müssen, dass es VII Theses, sieben Hauptstüfe sind, worüber der Kandidat brefragt, woraus er geprüft werden könne, und dass diese sieben Hauptstüfe Alles erschöpfen, was sich nur immer von Harmonie sagen lässt.
In short: These seven chords exhaust all that Harmony has to say.
And below, we see Vogler using Roman numerals to show V–I and IV–I progressions (in a chapter on cadence formulas and chord progressions):
Weber: Versuch einer geordeneten Theorie der Tonsetzkunst zum Selbstunterricht (1817–1821)
Less than two decades after Vogler's Handbuch, Weber publishes his Theory of Musical Composition. The following two images (from the famous English translation by Warner in 1846) should be clear enough, but the point is that here we see Weber distinguishing between uppercase and lowercase Roman numerals to show qualities of chords, which Vogler did not do.
In short, it was Vogler that began using Roman numerals. Vogler's original use was only for a single chord (VII), but by 1802 he was using Roman numerals for every scale degree. But it was Weber that eventually made the system popular. Whether it was through the practicality of his modifications (uppercase vs. lowercase numerals) or the success of his treatise, his system would go on to appear in treatises by Richter and Sechter, and even have influence in France, Britain, and even America.