Hi Amelia,
Your Davis is called a "New High Arm" or VF1 and were made with a few variations from approximately 1882-1893 in Watertown, NY and Dayton, Ohio. The Davis has no feed dogs as the underside photo shows and the material is advanced by the needle and helper foot . The owners manual shows some good pics of the shuttle, cabinets and use of attachments. I don't think you'll have any problems locating a shuttle.
http://www.ismacs.net/davis/manuals/davis-vf-manual.pdf
Good Luck
Jon