Question:

Can document fields that have default security set to Hidden be queried using the Vault API?

Answer:

The setting is found in: Admin --> Document Fields --> [Document Field] --> Security Overrides.  

If a Document Field's Default Security is set to Hidden and there are no Security Overrides then the field is not able to be queried using the Vault API. 

Related Documentation:

Vault Document Fields Documentation: Types of Field-Level Security