Some of these probate records are on line. Also some death records. St. Louis has more than most. Check out the Missouri Secretary of State web site:
http://www.sos.mo.gov/categories.asp?id=4I did a quick search and found several sullivans within your time frame. In particular there was a Daniel Sullivan who had a son named William, but his wife was named Nancy.
On the probate records, what you have to do is look through the papers to find where the heirs (i.e. widow and children) are listed. The records are not complete, and not everyone actually files for probate.
The secretary of state site also has some pre-1910 death information. I ran Mary Sullivan and got several hits.
Be sure you know exactly what city you are talking about. St. Louis the city seperated from St. Louis county (the suburbs) in 1876 and this makes a difference how you search.
Good hunting!