In the Wikipedia article on gravimetric analysis it is said that gravimetric analysis was used to determine the atomic masses of many elements in the periodic table to six figure accuracy. There are, however, no references to that claim.
Could someone please tell me, the molar masses of which elements can be determined using gravimetric analysis and what the exact procedure is?
Well, interesting question, I would re-word your question as "How can one determine an atomic mass with high accuracy using gravimetric analysis? Precision simply implies that you repeated a measurement and it agrees very well with itself even though it may be wrong. What we desire is high precision and high accuracy.
Almost all the naturally occurring elements had their atomic masses (used synonymously as atomic weight in older books) determined by gravimetry. For that you need to read Richard Theodore's Nobel Prize winning work who meticulously measured atomic weights for many many elements using gravimetry at Harvard in the early 1900s.
Do you know what was the experimental definition of atomic mass which made gravimetry possible? It is weight of an element which combined with 1 part of hydrogen or 16 parts of oxygen (we have to choose one of these but not both). Very sadly, nobody teaches this today so most chemistry students are oblivious to it and yet they can regurgitate the 1/12 of C-12 definition. This definition was invented for some other reasons rather than experimentally measuring atomic weights in the 1960s to resolve disputes among physicists and chemists.
So if I assign a weight of 16 units to oxygen and 65.4 parts of zinc combine with one oxygen in a 1:1 ratio (ZnO), we would assign an atomic weight of Zn as 65.4. Arbitrary as it may sound, this was the experimental way. What Richard did is that he used precipitation reactions using silver chloride and correlated that with oxygen 16.
I provide you with an exercise from Pauling's General Chemistry (forget about moles, Avogadro's, g/mol, molar mass etc.) to solve this problem. Let us go to 1940s, we have the above definition that atomic weight is weight of an element which combines with 16 parts of oxygen.
In a determination of the atomic weight of iron, 7.59712 g of carefully purified ferric oxide, $\ce{Fe2O3}$ , was reduced by heating in a stream of hydrogen, and found to yield 5.31364 g of metallic iron. Given the atomic weight of oxygen as 15.9994, to what value of the atomic weight of iron does this result lead?
Hint: Try to find what weight of iron would combine with 15.9994 parts of oxygen and note that there are two iron atoms and three oxygen atoms.
