Thanks for sharing the example. I had previously checked a 2020 census file (from NHGIS dataset 2020_PL94171), and in that case, the CBSAA field contains the correct codes. I’ve now checked tract-level files from several other datasets, and it appears that, generally, tract-level files from decennial censuses (at least 1990-2020) do contain metro area codes, but tract-level files from ACS datasets do not contain metro area codes. I also confirmed that this omission is original to the source: the base ACS summary files do not include this information.
NHGIS hasn’t added any codes to the files that weren’t provided in the original files, so unfortunately, you can’t get a direct tract-to-MSA crosswalk from NHGIS files. Instead, I’d recommend using the state and county codes that are available in the tract file to link to the Census Bureau’s CBSA delineation files, which contain corresponding state and county codes.