Scraping web content as a “logged in” connection

I have a subscription to a web site with data I want to analyze in R. While logged into the site with my browser, I can see the full table of the data that I want displayed in my browser. But if I'm not logged in to the site, it instead displays a subset of the data as a sample.

Though I am logged into the site with my web browsers, my R script, which uses base R's readLines(), behaves as though it's unauthorized (i.e., it returns the smaller subset of the data I require).

Is there a way to force readLines() to use my existing, logged in connection? Is there a better way I should be thinking about this?