Are there any kind of mathematical discoveries (proofs, theorems,...) that were made or supported by savants?
According to Wikipedia, Zacharias Dase "calculated a 7-digit logarithm table and extended a table of integer factorizations from 7,000,000 to 10,000,000." It also says he "calculated $\pi$ to 200 decimal places in his head, a record for the time," but it's not clear to me whether that was a record for calculating $\pi$, or just a record for calculating $\pi$ in one's head. I don't know whether Dase qualifies as a savant.
EDIT: Perhaps I should have mentioned that Dase did this in the 1830s or thereabouts, hence, without mechanical aids to computation.
I don't know if he qualifies as a savant, but Simon Plouffe (of the Bailey–Borwein–Plouffe formula) has also held a world record in reciting the digits of $\pi$.
Although not mathematical, but worth to mention:
Mental calculators were in great demand in research centers such as CERN before the advent of modern electronic calculators and computers. See, for instance, the 1983 book The Great Mental Calculators, whose introduction was written by Hans Eberstark.
from Mental calculator