Update staging table records in SQL Server 2017 with INSERT trigger and table-valued function, using INSERTED records
With Scott Hodgin's help (see comment thread above), I was able to figure out the issue:

First issue:  Since I certainly was using an OLE DB destination, I needed to add FIRE_TRIGGERS in the Advanced Editor (Microsoft, make this easier to find)

Second issue: once the INSERT trigger was firing, it was updating every record in the table, not just the ones where the function returned a value.  Of course it was; there was no WHERE clause or INNER JOIN to limit what needed to be update.  I modified my INSERT trigger to include the fields from the INSERTED table, and added a WHERE clause as follows:

CREATE TRIGGER [Staging].[INSERT_New_Pricing_tg]
    ON [Staging].[OrderDetail01_tb]
    AFTER INSERT
    AS 
    BEGIN
        SET NOCOUNT ON

        UPDATE Staging.OrderDetail01_tb
            SET NewListCost = A.CustomerPrice,
                NewPriceVersion = A.NewPriceVersion
            FROM (SELECT I.Company,
                            I.CustomerNumber,
                            I.UPC,
                            LU.CustomerPrice,
                            LU.NewPriceVersion
                    FROM inserted I
                    CROSS APPLY (SELECT * FROM dbo.Find_New_Price_fn(I.Company, I.CustomerNumber, I.UPC)) LU) A
            WHERE Company = A.Company
                AND CustomerNumber = A.CustomerNumber
                AND UPC = A.UPC
    END


Thank you very much Scott for starting me on the right path.  Without your help and that link, I never would have solved this.