This topic came up in a forum recently so this is my experience.
I once wrote a poem as a picture book, I did my own illustrations (I am an artist of sorts) and in the past some authors could be lucky and combine the two. But is is very unlikely unless you know the ropes, are known to a publisher and/or are a genius which I am not!
Illustration is an art and each illustrator has their own style. The pictures are intended not just to illustrate the story but add extra life, they sometimes add little additions or quirks that make it more fun.
My book jumped through a lot of hoops with Oxford University Press which took several months and they even seriously considered my preliminary illustrations. The two here are with the text added later.. In the end they wanted an illustrator of their own but by the time I agreed the person I was dealing with had left!
The moral of this story is to keep up to date, know the business, research current children’s picture books, not necessarily the ones you or your children once liked.
Picture books are expensive to produce and in rhyme format less likely to sell abroad due to translation problems, this is a big consideration for many publishers.
Self publishing If you self publish things in colour even with Lulu the copies will be really expensive, I did this with a collection of poems but it was for family and friends. You could as I did later, do the collection with b/w drawings which did sell and still does!
Diversify You could also use a colour illustrations and use the poem as a card. You could try and make it into a film and put it on youtube one of mine is called ‘The old dry stone wall’
Some publishers may be interested in a children’s prose poem but be warned they have there own in house illustrators.
This is a jumble of sketches in different media. I have used two of them for my poems can you guess which ones? I love drawing, painting etc. so can’t give up even if it never goes anywhere!
Other options A friend of mine makes gorgeous books for friends and her grandchild with handmade paper using various mediums to illustrate them. She binds them in different ways using raffia.
My mother made books for my children using ordinary scrap books. She recorded their adventures in bold felt pen, often with rhythm but not in rhyme. She pasted in tickets, cut out pictures, the children’s own drawings, feathers and bit and bobs collected during the course of that particular day. The children loved them, the reminder of the trip, the family featured in their own book and because they knew the story it helped their reading skills.
Writing and art is for many of us is simply a pleasure. (the answer to the sketches was 4 and 6)
The old dry stone wall on youtube