First thought: make a linked record field between your “products” and “requests” base. That way, they can select multiple products via the form if you add that field (which is in the “requests” table I imagine). The e-mails to the managers could then be done via the automations (eg via the “find” action).
So they click on the button in 1 record to open the form and then can select some more. The product they clicked on could be already filled via a prefill.