Yikes!! The "S" word. I refer to them as "creatures" b/c it makes me nervous if I say the word! I can't even look at photographs of them, so I feel your pain! Charlotte's Web was a scary movie to me - all those children!! Aaarrrgh!
My dad, loved to see creatures spinning a big, wide, web. He had the big garden that produced, eggplants, several types of tomatoes, 4 kinds of squash, and several varieties of peas, beans, potatoes, corn, as well as okra and peanuts, … sheesh, there was so much…anyway, Daddy always told me how good "creatures" are for the plant world, as they eat other harmful bugs. So, I guess there's that... but when you are truly scared of them, it makes no difference if they actually spin gold.
Trish, sadly there will be no way to rid your yard of them unless you are allowed to have chickens. Seriously. Chickens eat bugs. You can see them all over in Hawaii. They aren't kept for the eggs; they are natural predators of bugs of all sorts. You just need the little bandies, not the big . But I'm thinking you can't have those…so…think of natural, un-scary, predators. Birds, frogs, lizards, turtles, etc.
If you spray for them, you will be spraying toxins into your child's environment. Also, you know how tippy-toe those creatures walk anyway, so they don't get much of the toxin on them. In other words they are hard to kill. But your child will be crawling all over her play equipment so I would advise against any sprays.
You'll need to sweep out the corners of the little play house every day. Every day. Frankly, I'd remove the enclosed part and have it replaced with one that has a little tent-topper. That way the creatures don't have as many hidey-holes but your child has some shade.
You can also caulk &/or tape closed lots of little nooks and crannies that will be home to not only creatures but also bees!
Good luck with your new home and if you need someone who understands…feel free to contact me.
Meri