Your best bet is to contact the BLM Elko office first. You will want to speak to their records officer or whoever manages their records. They can be reached by email at email@example.com or phone: 775-753-0200. They might have a log for grazing permits over time. I would suggest that you work from the most recent person/company (if this is not protected by Privacy Law) that owns the grazing rights for this parcel of land backwards through the historical file. You will need to supply the BLM office with the legal land description for the parcel.
Please do not be surprised if they refer you to the Nevada State Records Officer. They can be reached by email at firstname.lastname@example.org or by phone at 775-861-6500. Most likely the older grazing files will be at Federal Record Centers either in San Francisco or Riverside, CA, which are part of the NARA system. To gain access to these records you will need to work with the Records Officer.
Finally there are older grazing files held by the Archival unites mostly at the National Archives in San Francisco and Riverside. A listing of the records can be found here in the National Archives catalog. Please review the catalog and reach out directly to the offices listed.