I often get letters from people asking me to illustrate a story they have written. Unfortunately, this is not possible since I'm always too busy with my own projects.
Most illustrators can't take on speculative work for individuals without a contract from a publisher or self publisher.
In most cases, publishers want to choose the artist for any given manuscript... so it's best for the author to submit stories directly to a publisher first.
What's the best way to go about submitting a story to a publisher? The first place to look for information about this process is the SCBWI.
Society of Children's Book Writers and Illustrators
Another good place for information is the Children's Book Council