Determine the market that best suits your children's book. Narrow down the age group that the book is most suitable for, as well as the genre it best fits into, such as fantasy, humor, etc.
Utilize Internet search engines and books such as the "Writer's Market" and "Children's Writers and Illustrators Market" (both of which are published yearly) to find literary agents and publishers that market or publish the type of book you have written. You should also look at other books that are similar to yours and note who has published them. You can do this at book stores and libraries. Note that may publishers will not accept unsolicited manuscripts unless they are sent by an agent.
Prepare a query letter to send to agents and/or publishers. The query letter should be professional, descriptive and to the point. You query letter should begin by stating the basic information of the book such as the approximate length, gender and age group it is targeting and the title. Next, explain the plot and characters. Don't be afraid to "give away" the ending. Publishers and agents want to know what their dealing with. Move on to list any publishing credits, awards or other reasons, such as degrees or work in child care, that you believe qualify you to write children's books. Explain why you feel your book will be successful, what books it may be similar to and how it is different from other works. Finally, be sure to thank the agent or publisher for their time.
Tailor each submission that you send out to the particular agent or publisher that you are sending it to. Be sure to read their submission guidelines and follow them to the letter. It is best to specifically address an individual in your query so that it doesn't feel like a form letter that you are sending to multiple people. Some agents and publishers will state that they do not accept submissions that have been sent to multiple recipients, but that should not dissuade you from sending multiple queries out. Be sure that your contact information is clear and use every opportunity to explain the less obvious reasons that you are qualified as a children's book author, such as the fact that you are a parent, work with children, read children's books with your children or nieces and nephews, etc.
The final part of submitting a children's story, and the most difficult, is waiting for an answer. Be prepared for rejection. Rejection is not necessarily a reflection on the quality of your work. It can be very difficult to get work published, and some great works and authors have received multiple, form rejections before being successful.
Work on your next project while you are waiting to hear back on your submissions. If you feel that you will not be successful with traditional publishing, and you really want to get your work out there, consider self-publishing. There are a number of good print-on-demand services that can help you publish your work, though you should be prepared to pay for the services.