You won’t be able to use any of the data-sources functions because the concept_hiearchy is used to provide the hierarchy of concepts on the domain type reports.
Otherwise, if you don’t use DataSources you should be able to start WebAPI without those tables.
However, I have found that the achilles tables do need to be populated due to the ‘record count’ functionality that is provided in the vocabulary search functions, so you should try to at least create the achilles tables in the CDM source your WebAPI references.