Apple - Is it possible to search for text in iBooks or the Kindle app?

Yes, in both the iBooks and Kindle apps, you can search the full text (excluding text in images) of the book you are currently reading. Here's how to do that in each app:

iBooks app

While reading a book, tap the screen. A magnifying glass icon becomes visible in the top right, along with other controls. Clicking the magnifying glass icon brings up a search box. After entering your search term in the box, iBooks presents a list of results from the current book, underneath the search box.

Kindle app

Similarly, while reading a book, tap the screen. At the bottom a magnifying glass icon will become visible. It works much like the iBooks search feature.

What about searching across all books?

If you wanted to know whether you can enter a search term at the top-level of these apps and have them search across all books stored in the app: No, currently neither iBooks nor the Kindle app have such a feature.

However, I have seen this capability on the Kindle device itself – I own a Kindle keyboard model and there is a "search my items" option available that will search all books on the device for a given term.