Check with friends and family first. If you know someone who has the ability to illustrate, talk to him or her about working on your book with you. In many cases you can work out a partnership deal that includes a shared percentage of book royalties instead of an upfront payment. If you know someone you can work with professionally, this can be an ideal situation.
Visit the Society of Children's Book Writers and Illustrators website (see Resources). This is an organization offering three levels of membership, depending on your publication history. There is an option available to sign up if you are an unpublished author. When you become a member, you will have access to children's book illustrator portfolios. Browse through the work. If you find someone whose style you like, contact the artist with a proposal for your project.
Negotiate with the illustrator for the terms you would like. You have the option of buying the copyright to the illustrator's work outright, if the illustrator is agreeable to the terms. This means you will pay a one-time flat fee and own all rights to the illustrations. If you'd rather, you can offer a partnership in the book, with a 50/50 split of royalties between the illustrator and you. If you can't come to an agreement, you can look for another illustrator.
Hire a lawyer to draw up the contract agreement between you and the illustrator. Regardless of how well the communication is between the two of you, ownership of material and book publishing agreements is a legal matter that needs to be handled by a legal professional.