Your PGA Tour card can be earned after playing through the Web.com tour matches at the beginning of career mode. After completing the Web.com tour, you must place in the top 25 of the earnings list to earn your PGA Tour card.
I played through 8 total rounds from first starting the game before earning this. Setting each tournament to one round helps you play through the career faster as well. Don't fret if you have a couple bad rounds, I had three where I was around +20 while I was still learning the game (especially putting).