Only a subset of applications are available in the Amazon store. If the client you need is not there, enable Settings → My Fire TV → Developer Options → Install unknown apps.
Install Downloader, enter the direct link to the APK (the provider usually provides it in the instructions) and install the application. After the first launch, a request for access to files will appear - allow it.
To make the Alexa remote buttons work in the player, set up Shortcuts (for example, double-click Home → channel list). Fire OS 7+ supports profiles, so you can create a separate account for children with a limited list of applications.