I took a look around and found an early reference to base ball. In 1862 The Eddyville Star reported that:
The young men of Eddyville being desirous of promoting rational enjoyment, have, or are about to organize, under the above name (Union Base Ball Club). They, we are informed, play Wednesday and Saturdays, in Benedict Hall. Too much praise cannot be bestowed upon the young men comprising this club. We are happy to say that the names of some of our best citizens are already enrolled. (names listed)
|The Eddyville Star - April 19, 1862|
Looking at the list of Earliest Baseball Clubs from MLB's Baseball Memory Lab, I see that the first base ball club in Iowa is in 1864 in Dubuque.
Looking at Protoball's list of Pre-pro Clubs and Games in Iowa I see that they have a club listed in 1858 from Davenport.
So, Eddyville's Union Base Ball Club isn't the earliest in Iowa, but it is early.
Note: I digitally cleaned up some of the detritus on the article image, making it a bit easier to read.