You have to stop worrying what other people will think. Your kids need to be able to have people over. I really think you should tell your son you changed your mind. They want to play with puppets they made!!! That's really adorable, and you need to support that.
When the parents drop their kids off they will probably chat briefly, and then leave. You should be able to do a few minutes of chit chat. You won't have to become their best friend.
You really need to push yourself to do this, not only for your son, but to expand your own world. It's time to become comfortable talking to other people. If you don't know what to say, ASK QUESTIONS.
Please let your son have his friend over. Who cares if your house is messy -- you have a lot of people in small quarters, so you have a good excuse. Kids who grow up unable to bring friends over are inevitably hurt by it.
Just quickly pick up the main room before someone shows up. Stuff things in a bedroom, or something.
ps -- when my kids were little, a friend of mind told me that the first time she ever came over to my house she told her husband afterward: "She's really nice, but she's a disaster," referring to the toys and stuff that was strewn all over the place. It's kind of funny looking back on that, because my house doesn't look like that at all now, but then I don't have little kids any more. Anyway, that woman became my best friend, so I guess she didn't mind my mess that much.