Search for the keyword "Selenium" at http://search.cpan.org and you will see that this tool is very well-supported by Perl. The selenium web-site also contains a lot of documentation that should be reviewed carefully.
While, depending entirely on the site, the task still might not be easy, Selenium might make it possible.
Perhaps, Selenium could be used to get far enough that the login sequence has been completed, and you can extract cookies or other tokens that could be used in the subsequent exchanges that could be driven by other Perl technologies. Unfortunately, there is no way to predict what might work best – or, at all.