How to use iPad without finger touching?

You can buy a stylus just about anywhere. Apple doesn't carry them, but virtually every electronics store on the planet does, not to mention Amazon, eBay, and most other online retailers that sell iPhone and iPad accessories.

Heck, you can go down to the grocery store or corner drug store when I live and buy one...

how do you mean 'not touching it'?

A stylus that moves by itself? Or something that would be the equivalent of a mouse? To manipulate the screen without actually touching it?

Because you can use a capicative stylus - the ones with the soft rubber tips - to manipulate the iPad's screen without your fingers touching the screen, but they have to touch the stylus.

Maybe some explanation would help folks come up with suggestions.

I have one that I bought a brookstone. It has what looks like a soft metal mesh head on it, and I have several that we bought for use where I work. They have a soft rubber tip and are used on iPads by nurses and aids who are typically wearing gloves. I've used both when wearing heavy gloves.

I hope you're aware of these features:

Using AssistiveTouch on your iOS device - Apple Support

The above article also references some accessories you can attach.

Yes. They work just fine. I have one (several actually) that I primarily use in the winter when I don't want to bother taking my gloves off and need to do something that Siri can't do for me.