And I can see exactly how they got to that mistake...

The writer was waffling between "We do not require customers to have a domain registration" and "We do not require that customers have a domain registration", and they forgot to remove the entirety of the old wording when replacing it with the new.

I've made this mistake myself several times, and it jumps out at me whenever I see it.

Very fitting. 'Customer Obsession' is Amazon's first leadership principle [1].

[1] http://www.amazon.jobs/principles

What's the problem? It seems totally fine to me, but I'm no English native.

As he says, there's an extra 'customers'. It should have been either

"We do not require that customers have a domain registration (...)"

or

"We do not require customers to have a domain registration (...)"