After searching around for a while, I came across Simon Grimm's video in which he added this exact plugin to an Ionic/Capacitor project. Prior to finding the video I was having trouble figuring out how to add the Android configuration properly to AndroidManifest.xml. When running ionic cap sync, I got a warning that I needed to add the following to AndroidManifest.xml:
<intent>
<action android:name="android.intent.action.SENDTO"/>
<data android:scheme="mailto"/>
</intent>
But there was no documentation on how exactly to do this. The solution was in the video:
<manifest ...>
<queries>
<intent>
<action android:name="android.intent.action.SENDTO"/>
<data android:scheme="mailto"/>
</intent>
</queries>
</manifest>
The intent element needs to be placed inside <queries>, which should be placed directly within the <manifest> tag (and not inside <application> or <activity>). After doing this, I was able to get the email composer to open up on Android 12 with Gmail.
Reference: https://www.youtube.com/watch?v=AF0jnmbbpbE