Finish the children's book before you try to find a literary agent. Also generate a list of additional ideas which you will be working on in the future. A literary agent will want to see that you have more than one idea for a children's book.
Research literary agents using a valid source of literary agents, including the Association of Authors' Representatives (AAR) and Preditors & Editors (see Resources).
Create a list of potential literary agents who work with children's book authors. Research information about the literary agent at their web site.
Prepare the children's book manuscript according to the literary agent's guidelines. Pay attention the specific details of how they would like to receive the manuscript. Send only the number of pages they request in their submission guidelines.
Write a query letter to the literary agent to accompany the manuscript. Inquire by phone or by email if the agency works with the Random House children's book division.