Finish your manuscript. Before sending out a single query, make sure your manuscript is as strong as it possibly can be. This means finding people to critique your work and revising until it shines. The Society of Children's Book Writers and Illustrators offers its members access to online critique groups. Alternatively, join a local children's writer group in your area to find people who will provide an objective opinion of your book.
Look at the acknowledgments section of your favorite children's books. Many authors thank their agents. If you find a name, write it down.
Look through books such as the most recent edition of the "Guide to Literary Agents." Check the sections for agents who represent children's writers. Add any names you find to your list.
Visit websites such as Agent Query and QueryTracker. These websites have search tools you can customize to create a list of children's book agents. Add appropriate agents to your list.
Evaluate each agent on your list. Look up their deals on Publishers Marketplace. Search for them on Preditors & Editors. Read their guidelines and bios to see if they sound like a good fit for your work. If an agent has a blog, begin reading it. Search online to find other writers' experiences with the agents. If you belong to SCBWI, discreetly ask other members for any inside info they have. Cross any agents from your list if they aren't legit.
Read each agent's guidelines. If an agent doesn't have a website, try Publishers Marketplace or the "Guide to Literary Agents".
Compose a query letter. The query should address the agent by name -- it's a business letter. Give the title, genre and word count for your book, then give a brief description of your story. End by mentioning any relevant credits or experience.
Send your query to the children's book literary agents on your list and wait for their responses.