I've having the exact some problem. Calling all community members, can we get some help pretty please?!
It's possible this is due to a certificate issue. The online installer is signed using a SHA256 certificate, which may not be compatible with certain versions of Windows Vista that do not support the latest signing technologies. You can try the offline installer posted at the bottom of the Installation problems | Flash Player | Windows 7 and earlier page in the 'Still having problems' section, however, there is no guarantee that it'll work.
Alternately, you can download and install an update from Microsoft from this https://support.microsoft.com/en-us/kb/2763674 page.