How do I exit the email app on an iPad with a home button?

The iPad User Guide is a rich and often overlooked source of information about iPad and iPadOS. Here is a direct link to the relevant section of the online Guide that describes switching between open Apps:

Switch between apps on iPad - Apple Support

This section of the Guide provides a comprehensive explanation of the main control gestures for iPad:

Learn gestures to navigate iPad - Apple Support

One method to exit the Mail [or any other] App, or switch to other open Apps, is to use the App Switcher page (also known as the Exposé view in iPadOS 26/27):

• View All Open Windows/Apps (Exposé): Swipe up from the bottom edge of the screen and pause in the middle before lifting your finger. This displays a "bird's-eye view" of all open applications and their associated windows. Tap the window view of the App to which you wish to switch.
• Switch Apps: In the Exposé view, swipe left or right to browse through the different apps. Tap the window you want to open.
• Quickly Switch Between Recently Used Apps: Swipe left or right along the very bottom edge of the screen to quickly jump to the last-used app without opening the full Exposé view.
• Close an App: In the Exposé view, a single app or group of apps can be closed by swiping up on the app preview/card. 

You can also easily return to the Home screen with an upward flick from the bottom edge of the screen.

That should have said without the home button