Hi Sarah--The book is called "We Help Mommy" by Jean Cushman, illustrated by Eloise Wilkin, originally c. 1959. I don't know if it is still in print; I got my copy new in the 1980's when my three youngest children were small, so it was still being reprinted at that time. The part you remember is Martha and Bobby's "put-away song" they sing as they're tidying up their bedroom; the rest of the story is not told in rhyme (the little girl, Martha, tells the story.) My copy is #305-52; Steve Santi's reference, Collecting Little Golden Books, 4th edition (there are newer editions of his book) shows a 1992 edition of "We Help Mommy". Hope this helps!
I remember my parents reading me this book as a child, I may have some of the last few pages mixed up, but does any of this sound familiar??:
In the box we put the blocks the cows (or dolls maybe) go in the crib, teddy bear sits on his chair, puts on his bright blue bib.
I'd really love to find this book.
Sarah