I don't know about any apps, but:
Do you use Google Calendars? And are you set up as an Exchange account so that you can display multiple calendars?
Google Mail as Exchange
Set Up Your Apple Device for Google Sync : Google Sync (ActiveSync) - Google Mobile Help
If yes, then you can do it, though it isn't pretty. Each phase will show up as an all day event, not a nice picture of the moon.
Open your Google calendar in a browser. On the left there is a section that is for Other Calendars. Click on the Add link just below and choose Browse Interesting Calendars. Select the More tab and subscribe to Phases of the Moon. Go back to the calendar and pick a color you like. {Surprisingly this time my colors on the iPad actually matched. They didn't used to.}
Now go to m.google.com/sync on your iPad, select your device and check the Phases of the Moon calendar to sync. Save.
That is all there is too it. When you open Calendar on the iPad it might take a minute or two for the new calendar to show up. It's just a text description of the phase and a time, but it's something.
I'm pretty sure it is possible with MobileMe too, but the Google calendar method looks like the better, easier, choice for setup. It all depends on being able to subscribe to a calendar, and there is no direct way of doing this on me.com.