There are a couple of ways to get the data you want. Since your question is asking specifically for coordinates and not actual GIS data (ie a shapefile), this question may be off topic / more appropriate at the Open Data StackExchange.
My first suggestion is the second link at http://www.stats.indiana.edu/maptools/ which will give you a coordinate list of all Indiana place names in an Excel spreadsheet. You will then need to filter the list to match a county seat list.
If you wanted GIS data, you could visit the Census website, Geospatial Data Gateway, or any number of other sites to download a point shapefile of all cities in Indiana, and again filter it to county seats. Some of those datasets may actually have that information as an attribute you can filter on rather than comparing to another list. You'd then have to add a couple of attributes and calculate the x,y coordinates.
If you just want the centers of the countys, or more properly centroids, you can download a county shapefile and use GIS software to create those points and again calculate x,y attributes.