This may be harder than you anticipate and PHP may not be the best language to achieve what you are after as it can not run on top of Outlook or any other desktop based email client. To do this you would have to run the PHP code from the command line, and have it set to regularly intercept all your emails and check to see if the email is from a specific user, and if it is parse the email for the two necessary links. Once you have the links you can have the code establish a cURL connection to the login form and pass the username and password, ensuring to pass any cookies defined from the authenticated session, and then establish a new cURL link to the download link to download the file to the local file system, after which you can send a new email to yourself attaching the file as a native attachment, dependant on file size.
Information on sending cookies with cURL can be found at How can I send cookies using PHP curl in addition to CURLOPT_COOKIEFILE?
The net result though is that this would have to run in the background continuously and would have to be set to regularly connect to your email server and check all emails for the existence of emails from the automated email sender.
An easier solution would be to find out if the automated tool can be setup to simply send the file as an email attachment so that you don't have to run the excess code. In addition any time the initial email structure changes or the download or login links change you would have to update the code to deal with the associated changes.
Main point is that PHP isn't the most ideal solution for what you are trying to do and what you are trying to do, in any language, is going to be a complex task to achieve.