Using planimeters to illustrate Green's theorem is a relatively recent didactic development. Neither Green, nor Cauchy, nor Riemann had any interest in the instruments, and vice versa, planimeter developers did not involve something so abstract in explaining their "principle" mathematically.
The kinds of mathematical proofs given can be seen in Shaw's Mechanical integrators, including the various forms of planimeters (1886). They were not formal, but then no proofs were before formal mathematics was introduced at the end of 19th century. The idea was to directly reduce the wheel movements to what we would now call a Riemann sum. One finds proofs of the same sort in modern textbooks. Shaw seems to take Amsler as his principal source on mathematics. Amsler was a Swiss mathematician who invented the polar planimeter in 1854, and was likely familiar with Gauss's work on differential geometry from his studies at Königsberg in 1840-s. He is explicitly credited by Shaw for extending the idea to non-flat surfaces:
"As the Amsler planimeter alone, so far as the author is aware, has been modified to measure the area of any nondevelopable surface, this modification may be here noticed... The theory of the action
of this instrument has been fully explained by Professor Amsler, in an article in which the theory of the relations between measurement upon a spherical surface and upon a plane surface is discussed."
But the more basic planar proofs are certainly older. Maxwell ventured into constructing his own planimeter ("platometer", as he called it after Sang) after seeing them displayed at the Great Exhibition of 1851 in the Crystal Palace, London, which generally gave them prominence. In Description of a New Form of Platometer an Instrument for Measuring the Areas of Plane Figures (1855) he notes that "a very able exposition of the principle of such instruments will be found in the article on Planimeters in the Reports of the Juries of the Great Exhibition, 1851", and gives a geometric argument presumably modeled on it. Here is an excerpt:
"Therefore, if we have a machine with an index of any kind, which, while the generating line moves one inch downwards, moves forward as many degrees as the generating line is inches long, and if the generating line be alternately moved an inch and altered in length, the index will mark the number of square inches swept over during the whole
operation. By the ordinary method of limits, it may be shown that, if these changes be made continuous instead of sudden, the index will still measure the area of the curve traced by the extremity of the generating line."
Earlier publications by the original inventors are hard to track, but one can expect arguments of the same sort there, perhaps without mentioning limits. The earliest known one is by Gonella of the University of Florence, who reinvented a wheel-and-cone planimeter in 1824 and published almost immediately. He also quickly came up with the wheel-and-disc design, which was the one he presented at the Great Exhibition of 1851.
Hermann, a Munich land surveyor, who invented the earliest known planimeter in 1814, not only did not publish anything, but even used it only privately. It is first described in Bauernfeind's Zur Geschichete der Planimeter (1855) published in Dingler’s Polytechnische Journal. There might have been earlier private inventions, now lost, and independent reinvention continued until 1851, when Sang invented his platometer, see Care, Illustrating the History of the Planimeter.