It has been my experience that I can most of the time find the really old obituaries by contacting the library or archives and ordering them. However, finding the ones that are either recent or not really that old- sometimes proves not that easy. Below are some links that I have found quite useful in obtaining obituaries. Many of these are from areas outside of Morgan County, but I have consistently found my ancestors obituaries in these areas. If when reading obituaries, you look near the end, you will find the mention of the funeral home and sometimes a link to it.
I should note that on some of these links, you may have to scroll to the bottom of the pages to obtain the links for the Obituary Archives. Under the FUNERAL HOME links- you must click on the link for Obituaries.
If anyone should have any other sources, please send them to firstname.lastname@example.org
West Parrish Pedigo Funeral Home -- Spencer, Owen County, IN
Jones Family Funeral Home -- Mooresville, Morgan County, IN
Moore Funeral Home -- Brazil, Clay County, IN
Neal and Summers Mortuary -- Martinsville, Morgan County, IN
Schoppenhorst Funeral Home -- Brazil, Clay County, IN
Welch & Cornett Funeral Home -- Linton & Worthington, Indiana
Meredith-Clark Funeral Home -- Morgan, Brown & Johnson County, Indiana
Carlisle-Branson Funeral Chapel -- Mooresville, Indiana
Costin Funeral Chapel -- Martinsville, Indiana
Light Memorial & Funeral Chapel -- Monrovia, Indiana