What is this disgusting orange spider?

That is one of the European garden spiders, Araneus diadematus species.

Sad you find them disgusting, they are far from that. They are beneficial, not dangerous at all. Check out that beautiful web. Without doubt, one of my favorite spiders, welcome at my place any time.

Used to play with them as a kid.

I see yours are staying out in the daytime now, typically, they eat their web with the bugs before hiding out for the day, come out at dusk and make that fancy web.

Serious now, I will gladly take any you don't want.

https://bugguide.net/node/view/3376

A couple videos for you:

http://www.youtube.com/watch?v=OraClk3gUO0

https://www.youtube.com/watch?v=86tp3asD9kM

Could this be it? https://www.youtube.com/watch?v=m-xTWMbuCwwumpkin spihttps://www.youtube.com/watch?v=m-xTWMbuCwwder.

Oh, he's cute. Please don't kill him!

Looks like a species of Orb Weaver. It may be easier to identify exactly if you gave your location.