Here are a few screencaps and other images of Éomer and Éowyn from The Two Towers, and from The Return Of The King. While most were capped by Jai, they are the property of the filmmaker and studio. Take them and use them if you like, just don't direct link.
Of all of them, this one is my favourite, as it says so much. Thanks to Taerwen for sending it to me. You can read Taerwen's thoughts on Éomer and Éowyn's relationship on the Relationship page.
If you have any pictures you'd like to donate, please contact me! Full credit given, of course.