BallparkDJ makes it very easy to share your team with another BallparkDJ user from the Actions Menu > Share Team. However, the process described below allows you to share your team with anyone, including Facebook or other social media. This process can also be used to extract the professional voicing introductions for external purposes.
iPhone now provides a convenient Screen Recording feature that captures both the screen and the audio into a video that you can share via email or upload to Facebook. Go to Settings > Control Center > Customize Controls, and click the green + button next to Screen Recording to include it on the Control Center screen. Now, go to Control Center and tap the Screen Recording button (looks like a circle/target). A countdown from 3 will precede the recording so you will need to toggle over to BallparkDJ quickly and begin playing your team intros. You can put your team on Continuous Play so that it plays all players sequentially (by clicking on Single Play button at lower right of team). Once completed, click on the red time at the top left of screen to stop the recording. The video will automatically be saved to Photos/Videos and may be emailed or uploaded to Facebook or other social media. It may take a few tries to get the timing just right.
If you intend to share the video on Facebook, you may consider removing the music before recording in order to avoid any violations of music copyrights. You can do so relatively easily without impacting existing team. Just choose Actions > Duplicate Team to create a duplicate of team. The edit each player (on duplicate team) to remove the music by choosing "No Music" in the Edit Music window.