Based on the error message you are geting, you have to change the alias of the columns. Here's an updated version of the query:
SELECT [Staff Database].[Last Name], [Staff Database].[First Name], [Staff Database].[Program Office],
IIf([DL Exp]>=[Start Date] And [DL Exp]<=[End Date],[DL Exp], NULL) AS
[DL Exp1],
IIf([Car Ins Exp]>=[Start Date] And [Car Ins Exp]<=[End Date],[Car Ins Exp], NULL) AS
[Car Ins Exp1],
IIf([Car Regis Exp]>=[Car Regis Exp] And [Car Regis Exp]<=[End Date],[Car Regis Exp], NULL) AS
[Car Regis Exp1],
IIf([Car Inspec Exp]>=[Start Date] And [Car Inspec Exp]<=[End Date],[Car Inspec Exp], NULL) AS
[Car Inspec Exp1]
FROM [Staff Database]
WHERE ([DL Exp] >= [Start Date] AND [DL Exp] <= [End Date]) OR
([Car Ins Exp] >= [Start Date] AND [Car Ins Exp] <= [End Date]) OR
([Car Regis Exp] >= [Start Date] AND [Car Regis Exp] <= [End Date]) OR
([Car Inspec Exp] >= [Start Date] AND [Car Inspec Exp] <= [End Date])
Regards,
SQL Server Helper