It’s not a silly question! Unfortunately, the geographic codes in the source ACS summary files from the Census Bureau are generally limited to the key identifying codes for a given level. Even though they certainly could identify the CBSA in which each block group is located, the CBSA field is left blank for block groups, and for all other levels that don’t include CBSAs in their hierarchy.
E.g., the “metropolitan division” level includes CBSAs as a parent level, so metropolitan division data will include CBSA codes. In county data, there are state codes because states are in the county hierarchy, but there are no CBSA codes, nor region or division codes, even though counties nest within those areas, too.
FWIW, the geographic info in decennial census summary files is more complete, providing codes for all containing units.
To get CBSA codes for ACS block group data, I think the most straightforward approach is to use the Census Bureau’s CBSA delineation files, joining them to the block group data based on the state and county identifiers.