Hello 1230carondelet-,
We understand that you're having an issue with your iPhone 13 not responding and not being able to retrieve mail. We'd be happy to provide you with some information that can help with what you've been experiencing.
First, if the device is frozen or unresponsive, you'll want to try the steps in the page below to resolve this issue.
1. Press and quickly release the volume up button.
2. Press and quickly release the volume down button.
3. Press and hold the side button until you see the Apple logo.
4. If your phone doesn't turn on, follow these steps to check your hardware and charge your phone. You might need to charge for up to an hour.
5. If your phone still doesn't turn on, contact Apple Support.
If your iPhone won't turn on or is frozen
Once you've gotten the device to become responsive, you can go through the steps in the two resources below to help with the Mail app and not being able to retrieve your emails.
If an app on your iPhone or iPad stops responding, closes unexpectedly, or won’t open
If you can't receive email on your iPhone, iPad, or iPod touch
Thanks for using Apple Support Communities.