In theory, the application should work fine under Windows 10, even though the compatibility isn't listed on the developer's website. However, in case you run into trouble, you can simply right-click the executable file and go to Properties. In the window that opens up go to the Compatibility tab, check the box marked Run this program in compatibility mode for, then select an older version of Windows like 7 or XP and press Apply.
If that doesn't work either, you can set up a virtual machine, install an older version of Windows on it, then run KODAK Share Button App from there.