I'm not sure of a way to do it with just the Shopify notifications, but you can do it with the app Order Automator. You can, for example, create a rule to send an email notification to someone@yourteam.com if the order has a product of type A, or tag B, etc.
My team built the app so if you find you need a customization outside of what's available feel free to give a shout, we do customizations.
Founder of
Speed Boostr (Shopify optimization experts, theme customization, custom app development).
Creator of
Shopify Analyzer (first performance analysis tool for Shopify sites, free for the Shopify community).
Creator of
Order Automator (auto tag orders and customers + auto fulfill orders + email notifications).
Creator of
Theme Scientist (A/B testing app) and
Tip Jar (add a tip button to your store).
