I can offer a few observations and suggestions as a member of our community with a degree in paper science and engineering and having spent 40+ years in an around the industry...
Seems like it would be easy-peasy to use a compass and either scissors or a sharp blade to cut the ring shims you need.
Paper is pretty resistant to oil. It's water that breaks the bonds holding it together. And even those bonds can be made quite durable by using wet strength grades. Think packaging like beverage carrier board (typically 0.020", more or less). There are many other wet-strength grades -- labels, coffee filters, some paper towels, to name a few -- but the paperboard grades are probably most like what you'd need. A grocery paper bag could be a good thinner caliper wet strength stock (probably 0.012", more or less).
Even without being a wet-strength grade, I'd still think solid paper or paperboard (not corrugated). Better to use brown or white rather than recycled gray for strength.
Something like a business card or pharma packaging might be good lighter grades (say 0.010"). Office printing/writing grades, such as a copier paper as noted, could be a good thin stock (probably around 0.005").
I'd saturate a non-wet strength paper with oil to give it some resistance to water or humidity.
Gasket papers are available as mentioned. McMaster-Carr has a selection. They're almost always wet strength grades and often saturated with rubbery latex resins. I don't think you really need that for what you're trying to do.
I’ll knock up a few from some paper, certified grocery bag sounds like perfect material! was thinking I’d be able to get a few for a few cents somewhere, certainly not paying $15 for 10 little rounds of paper. Appreciate the paper experts weighting in to help 
