Vudu isn't available on Fire TV/Stick.
True, not available in their app store but it can be side-loaded like many other Android apps. You can use an app called "Apps2Fire" or you can download the Vudu apk to your Dropbox account and install to the FireTV using ESFile Explorer app. But yes, you can install Vudu onto the FireTV or other Android based streaming devices. But it is easier and more fun to have 2 streaming devices on your TV. I myself, have a Roku plus another companion streaming device per TV. Other streamers to include GoogleTV, Sony Blue Ray, Google Nexus Player, Tivo, Equiso Streaming stick and first gen Fire stick.
Hope this information is helpful.
2xRoku2 XS,Roku1+SE,Logitech Revue GoogleTV,Raspberry Pi+XBMC,Mac Mini+EyeTV DVR+RabbitTV+PLEX+Kodi,TiVo Premiere,Sony BluRay+streaming apps,Equiso Android TV,FireTV,Nexus player,Simple.TV Local TV:QAM from ISP coax line. PLEX+unsupported apps