I am pretty sure alansidman's method will work. My method is a VBA solution where his is a query solution. My code would be put in a button or maybe a "lost focus" event if you want it to be triggered as soon as the object has lost focus. I recommend putting it in a button if only for testing. Below is a more complete set of code that you should put in the click event of your button:
Code:
Dim vResult as Date
Dim vDriverID as String
Dim mySQL as String
vResult = DLookup("FieldYouWantToReturn", "TableFieldIsLocatedIn", "FieldYouWantToMatch = DriverID")
mySQL = "UPDATE [table_name] SET [column_name] = #" & vResult & "# WHERE [other_column]='DriverID';"
CurrentDb.Execute mySQL, dbFailOnError
This will look up the Date you want and update the record in the new/desired table.