BillRecordID is Foreign Key to the Bill table's primary key RecordID. Vendor is a required field in the Bill table.
To execute this you'll need a few separate calls and code to connect the returned datasets
1. Select api call to locate all Bills where vendor equals yourVendor.
2. Query returned dataset for list of RecordIDs
3. Select call to appropriate BillLine Table where BillRecordID is in list of RecordIDs
4. Add the second dataset to first setting keys as appropriate.
5. Query in memory dataset as needed
HTH