I tweeted that you couldn’t do this, and got replies from a dozen or more people with ways to do it. And I’m happy to report at least one of them works.
1. Use the email address that Amazon provides for your Kindle. It’s on the Amazon site, poke around, you’ll find it.
2. The Kindle app shows up under Apps in iTunes. You can drag .mobi files into its Documents directory and they show up in the Kindle app.
3. You can use Dropbox. It will report it can’t open the files but if you look in the upper right corner of the screen there’s a way to send them to a bunch of apps, for .mobi files this includes the Kindle app.
There may be other methods, if you know of them, please add a comment below.