How can I set it up so when I order a product, I can send the order email to multiple people at the vendor?

When using the "primary contact for account" option in the trigger, it will only send it to the one primary contact for each company that it's sent to (and you can only have one primary contact for each company). But you could tag the people you want to send this specific thing to and then "use tags for recipient" when creating the trigger! 

Or you could use the new "custom email address" option instead! Then you can manually add several email addresses as recipients. But remember that means you will probably have to have a different trigger for each vendor...

custom email