A field in Bill of Lading::CarrierID should be formatted with this value list. The name field should not exist in the Bill of Lading table at all.
To show the vendor name from field two in the value list is done through one of two approaches:
1) Format Bill of Lading::carrierID as a pop up menu with this value list. It will then display the selected vendor name even when the focus is not in this field.
2) If you have a relationship in place to support this (and you should), you add the VendorName field from the related table to your layout to show the name. Some developers place this field on top of the ID field with behavior options set to block Browse Mode access to the field. When you click on the name field, the drop down list formatted ID hidden behind it pops to the front and deploys the value list. When you select a value, the field disappears back behind the name field and the name field update to show the name of the selected vendor.
Customers have a choice to use the pop up, (in this example already works) or bring it from another table as a memory variable... choose the carrier based on a quote.
The issue is when i set the field in the script from the memory variable. It should see the value list and set the ID in the value list, correct.
My experience with the dual fields is that it does not work well in IWP.
If necessary, a text field can use an auto-enter setting to look up the related table's name, but I doubt that will help much. IWP is technology no longer supported by FileMaker and it had a lot of limitations in what it could or could not do. In IWP it's often necessary to commit records before the view will update to show the correct data. You might even find it better to work from a selection portal instead of a value list as you can then make the user's action of selecting a value run a script where you then have more control over what happens when the value is selected.
Please make sure to let people know when you are posting a question for something that needs to work via IWP and a web browser or you may get help that isn't helpful as it can be a suggestion that doesn't work in your context.