Philosophical galvanism (as opposed to the empirical study of electricity in animal tissues) was a peculiar form of vitalism. While vitalism in general was quite popular and influential in the late 18th-19th century, this particular variation was not. Galvani himself soon abandoned references to élan vital, and after Volta's pile vitalists mostly did not associate it with something so specific. After the invention of the Voltaic pile in 1799, galvanism quickly lost currency with scientists, but remained in the popular culture for a couple more decades, albeit faded into the background. Mary Shelley's Frankenstein mentions it, so does Hegel, as late as 1830, see Burbidge, Hegel on Galvanism. Schopenhauer in The World as Will and Representation (1818/19), even tries to merge Volta's theory with galvanism and assimilate it into his scheme:
"The same thing shows itself in the lowest grades... when galvanism overcomes chemical affinity, decomposes
the closest combinations, and so entirely suspends the laws of
chemistry that the acid of a decomposed salt at the negative
pole must pass to the positive pole without combining with the
alkalies through which it goes on its way, or turning red the
litmus paper that touches it... For as every body must be regarded as the manifestation of a
will, and as will necessarily expresses itself as a struggle, the original condition of every world that is formed into a globe
cannot be rest, but motion, a striving forward in boundless space
without rest and without end".
But even this is a remark in passing, among multiple other examples of the Will "striving". Other than Schelling, perhaps the most prominent proponent of philosophical galvanism was Erasmus Darwin, a philosophizing physician, early evolutionist and the grandfather of Charles, see Erasmus Darwin, Galvanism, and the Principle of Life. Zoonomia, or the Laws of Organic Life (1790), where his speculative evolution theory was laid out, made Darwin famous. There he talks of "similitude between the spirit of animation, which contracts the muscular fibres, and the electric fluid" and soars to the élan vital driven evolutionary speculation, "that all warm-blooded animals have arisen from one living filament, which the Great First Cause endued with animality, with the power of acquiring new parts... and of delivering down those improvements by generation to its posterity".
Darwin inspired many, Schelling himself praised his vitalist take on galvanism in Zoonomia, contrasting it to "mechanical" explanations. Among others was Mary Shelley. The preface to the first edition of Frankenstein (1818) opens with "the event on which this fiction is founded has been supposed by Dr. Darwin and some of the physiological writers of Germany as not of impossible occurence".
In Germany, the soil was particularly fertile for vitalistic Naturphilosophie, given its congeniality with philosophical and cultural romanticism prevalent at the time. It influenced the "experiential science" that emphasized holistic and qualitative aspects over Newtonian mathematized mechanics. Pfaff, for example, tells us in Der Elektro-Magnetismus (1824):
"A physical explanation penetrates further than a so-to-speak mathematical explanation, which gives only a formula for the quantitative determination of the phenomena. It seeks to represent the phenomena in their larger general connections with the whole of nature and to connect the fact with which the
mathematical construction starts still higher with the essence of
the forces of nature themselves and thus to give an account of the
qualitative [aspects] of the phenomena".
One can see where Goethe's and Hegel's quarrel with Newton comes from. Schelling took his science from another "experiential" scientist, Ritter, see Schelling and experiential science by Breidbach:
"His terminology describing the potentialities and polarities of nature was formed during Schelling's collaboration with the physicist Johann Wilhelm Ritter. This scientist adopted the schema Schelling had developed for the categorization of natural phenomena to describe the peculiar facts that interested him in his area of research. Thus Ritter was able to develop a classification of the various phenomena of animal galvanism. Thus it can be shown that the idealistic "Naturphilosophie" was part of the scientific culture of about 1800. It is to be interpreted as philosophy of science and has to be evaluated not only in a philosophically systematic way but in particular in its influence on the way scientific categories were ordered at the time. Thereby it can be shown that the idealistic vocabulary had close correspondence to French morphology and English Natural Theology."
For more on the said culture, and its soon to come generational abandonment, see Caneva, From Galvanism to Electrodynamics:
"The abstractness of the physics of the younger generation contrasts
sharply with the qualitative Anschaulichkeit of concretizing science.
Against the former notion that one of the tasks of science was to
capture the essence of the phenomena, the new physics dropped the
requirement that a theory should provide a true representation of
physical reality [...] It is less important that men like Neumann, Fechner, and Liebig eventually turned from Naturphilosophie - and only Liebig spoke contemptuously of it later on - than that the younger generation as a whole felt no sympathy toward the way science was pursued by the members of the older generation of concretizing scientists and declined to take them as models. The failure of the latters' teaching to meet their students' expectations encouraged the younger men to forge their own model of science."