Here are a few things for you to try:
- Make sure that your iPhone Kindle Reading app is registered with the correct Amazon account.
- Verify that your Kindle Reading app is up-to-date.
- Ensure that Whispersync is enabled on your account and on each app, under settings.
- Restart your device.
You can manually send the ebook to your iPhone.
- Open Manage Your Content and Devices.
- Under Content, select Books.
- Find your book and on the right select “Deliver or Remove from Device“