Lawn really does mean lawn. You definitely can not bring lawn chairs in. I hope this link will get you to the FAQs at summerfest.com http://www.summerfest.com/event-info/faq
You will see that not much is allowed to be carried in. Many people bring in a small blanket or beach towel to at least cover the ground. The lawn seating is at the top of a fairly steep hill, so it would not really be practical to sit on lawn chairs anyway. You will not have an assigned seat in the lawn section. It is first come first served. None of the seating up there is real great, so there is no reason in my mind to rush into the amphitheater real early. I would enjoy the food and music on the grounds and go in maybe 45 minutes to an an hour before show time.
I suggest you take some time to read more information on the website to make your Summerfest experience more enjoyable. That includes info on getting to the grounds. Parking can be expensive and a real pain. Shuttles or buses are much better options. Summerfest can be a great experience, but it is large and can get crowded, so it can also be overwhelming.