The Internet Explorer app on the Xbox 360 does not support browser plugins such as Adobe Flash Player, Microsoft Silverlight or Java, as of January 2015. It is impossible to download and install Adobe Flash Player on an unmodified Xbox 360.
The Xbox Support page does not indicate that support for browser plugins is forthcoming. One potential work-around is to view the mobile version of a page, if this is supported by the website. To access the mobile site, users should press Y on the controller to display the web hub and then select the settings icon. In the settings, the user should make sure the check box beside "Use Web pages formatted for mobile devices" is activated and then close the settings page. Reloading the Web page may load an alternate format of the content that is compatible with Internet Explorer for Xbox 360.