How can I find my grandmother's OLD instrument rating documentation?
The FAA has an airmen inquiry search portal. I put Nathalie Brown and came up with Nathalie Myrtle Brown with the following details:

Certificate: PRIVATE PILOT   Date of Issue: 11/25/1968
Ratings: PRIVATE PILOT
AIRPLANE SINGLE ENGINE LAND
AIRPLANE MULTIENGINE LAND
INSTRUMENT AIRPLANE

As @FredLarson points out in comments the date supplied seems to be the last change to their database, i.e. address, name or addition of rating, so that means she must have gotten all of her certificates at or before the 25th of November, 1968.
This is a start, but it doesn't give you real dates. If the FAA has that much information they should have a lot more, to get details you should contact the airmen certification branch. I'm not sure what hoops you have to go through, maybe prove you are family, etc.
There is a big database of pilots from the FAA search portal, but it is not complete - your grandmother wasn't in it for example, and there's no dates of ratings anyway. Maybe the FAA can get you access to a better airman database with more of those records, it's possible they simply don't exist anymore though, you'd have to ask.
If she has personal effects left then her pilots certificate or other details may be there, certificates used to be paper, probably folded up. Really useful would be her logbook, which would be a record of every flight she made, there would be endorsements in these which would be useful in learning when she achieved her ratings.