They can. There is no age limit on this, it is judged on the quality. However, as a general rule, quality tends to come with age. :)
You should send a manuscript around to agents first. You can also send them to the publishers dirrectly then get an agent.
Random House and Scholastic Corp. are the two largest children's book publishers.
The most common children's book publishers is Peace Hill Press. You can see a list of the publishers at http://en.wikipedia.org/wiki/List_of_publishers_of_children%27s_books
You can send copies of your book to literary agents, publishers, book reviewers, and influencers in your genre. It's also a good idea to send copies to potential endorsers and those who may help promote your book.
There is a comprehensive list of publishers aas well as information on becoming a published Children's book author located at www.signaleader.com. There is also information on finding an illustrator.
Send your book out to as many publishers as you can and be prepared for rejection - don't let it hold you back, just keep on trying. :)
John Newbery was a publisher who is considered the "Father of Children's Literature." He published over 100 children's books when other publishers had not even considered the children's book market. His first book was A Little Pretty Pocket Book.
Bookspot.com lists Scholastic as the top children's publisher. They're followed by Random House and Harper-Collins.
You can publish your children's book through www.cafepress.com or www.lulu.com. These are considered "self publishers".
There are several websites that give information about publishers of childrens books. One of these websites is childrensbooks.about.com.
1) Make sure your book is thoroughly polished and you have read through it many times. 2) Write a shortened summary of your story (called a Synopsis) to pitch to the publisher 3) Write a query letter to put your book across to the publisher and catch their attention 4) Before you send out your query letter do some research on publishers and agents. Target publishers that publish writers with a similar style to you. 5) Send out your query letters to the publishers and send them back exactly what they ask for. 6) Don't lose faith as you will inevitably get rejected at least once. Keep writing and you will succeed.
One of the largest, if not the largest, publisher of children's story books is the Scholastic Book Publishing company. In the United Kingdom one of their largest children's book publishers is Penguin.