I have a table created with a Select Into ... statement. [...] Is there any way to have this table dynamically update itself (using the query that created it) whenever its data is accessed?
What you've described is a SQL VIEW, also called a "(saved) SELECT Query" in Access. A View is a virtual table that dynamically retrieves its information from other tables (or views) each time it is accessed. The view does not save its results between calls, so each time you reference it you get the most current data. See if you can use a VIEW (in SQL Server) or a saved SELECT Query (in Access) in place of the temporary table you are currently creating.