As of January 2015, the Xbox 360 Internet Explorer software does not support browser plugins such as Adobe Flash Player, Microsoft Silverlight, or Java. Adobe Flash Player cannot be downloaded or installed on an Xbox 360 that has not been changed.
Support for browser plugins does not appear to be on the horizon, according to the Xbox Support page. If the website supports it, one possible workaround is to see the mobile version of a page. Users should press Y on the controller to display the web hub, then pick the settings icon to access the mobile site.
The user should select the box next to “Use Web pages formatted for mobile devices” in the options and then dismiss the settings page. Reloading the page may result in the information being loaded in an alternate format that is compatible with Internet Explorer for Xbox 360.