They can but consider this: How would they know to send them?
If you are adding a provider only locally there's no way for the app to know to generate a notification if it doesn't have an IMAP idle connection open constantly or scheduled polling. If the app is shipped by the email provider (like gmail app for ios using a gmail account) they can send pushes without leaving connection open or polling.
The point of having your details with the app developer is that their servers will remain always connected to your email provierr and generate pushes that will go over Apple or Google's push network.