The report will need credentials to run. (Turning on anonymous access to SSRS is not supported and not recommended.)
Those credentials can come from a few different places. 1. Users are prompted for user name and password. 2. Credentials are stored in the browser (or in Windows.) As you've seen, this is easier to handle with Internet Explorer than Firefox for NTLM authentication. But Firefox does give you an option, as you've mentioned. 3. Some other service or website accesses the report server and hands in credentials. This other service then passes the report on to users. Designing this would require some thought: Would you need to track who accesses this service? How would you secure individual reports?
Your comment suggests that you have conflicting requirements: you aren't allowed to have Firefox automatically log in for security reasons, but you want to have Firefox automatically log in. There's no technical advice that can solve that problem.