Oh, NOW you tell us that your child has already been diagnosed with Sensory issues?
How helpful it would have been to inform us of that in your original question. How could you leave that out? You're outrageous.
If you think that home-schooling is the answer to a disobedient, rebellious child who refuses to act appropriately in public (assuming that said child has the intellectual capacity and mental resources to comprehend and obey), you are headed for disaster.
Homeschooling (and yes, I've homeschooled at times) requires discipline, both on the part of the child and the parent/teacher. Sure, there is a certain amount of freedom to select your own curriculum, or structure your homeschool in such a way that accommodates your family. Some people homeschool while living on a farm, and farm chores outside are part of the daily routine. Some people choose a fluid curriculum, where the child's interests dictate the curriculum. But regardless, a certain amount of discipline is required. There are several ways to arrive at a solution to a math problem, for example. Some homeschool parents choose Cuisinaire Rods, some go strictly by a textbook, some explore math in artistic or scientific or creative ways, but they all arrive at 2 + 2 = 4, not 2 + 2 = donkey or paint. Even if the point of the homeschool is to help the child direct the learning, there still are standards that must be met in order to fulfill legal requirements, and that means the parent has to be in charge.
Now, if your child cannot abide any clothing on her body, if wearing socks reduces her to tears, if she can't sit still for a minute without fidgeting painfully because of the weight, texture, or closeness of clothing, if clothing seams cause her anxiety and physical pain, then get her evaluated for sensory disorders immediately. If she is properly diagnosed as having a sensory disorder, there are clothing companies that make clothes for these kids, without seams, in particular fabrics, etc.
Also, have her medically evaluated. Perhaps she has an infection, or a parasite, or a skin condition in the area that underwear might touch, that makes underwear unbearable.
But if your child is wearing shoes and socks and shirts, and has no physical or emotional or sensory disorders, and she simply refuses to comply with perfectly reasonable standards, then you've got to step up.
No jungle gym when others are around? I'm assuming that means it's ok when the park is deserted. What you need to do is pretty much restrict all pleasurable activities. No friends over, no park or playground, no sitting in front of the tv or video game or computer, no tablet. Right now, she's the queen of the castle, and she is dictating your life. You don't have clear rules. No underwear is ok at some times, we'll try to disguise the fact that you're not wearing underwear by various means, mommy won't go to church because you demanded inappropriate attention...your child is living in chaos, and she's the master of it.