I didn't find any online rosters for the 106th when I did a search on Google, but you can get a roster if you use the Civil War Soldiers System website at http://www.itd.nps.gov/cwss/

You can do a search under Soldiers, then instead of entering a name of a soldier, enter the name of the unit. Or you can enter the name of a known soldier and when his information comes up, click on 106th Ohio Infantry. When the unit history comes up, scroll to the bottom, where you can opt to get the roster in an Excel format.

Note: this takes a while and it helps if you have a broadband Internet connection.

Hope this helps!