As far as we know, there's no real process for that - and least as it went for us, most of the deal between the sponsor and SE was done.
Now that there's rumours already started...
At time of this posting - there's no confirmation for or against anything going on with the AWS sponsorship of SU. We were told it was going to happen, raised a few points, and I believe they've been taken into consideration.
Firstly, I'm personally deeply disappointed that the assumption was made that this was cancelled. This could be potentially harmful to something that could help sites get a little bit of a higher profile with the powers that be at what seems a minimal cost/ change.
I'm also unsure if this would count as a strong objection - simply because the voting numbers were relatively small. We haven't been informed as such by staff, and the delays could be due to anything. I suspect if there was a change, that we'd be informed, and we have not been yet.
I'd also point out that, aside from my answer, setting out my views of this initiative– disagreement seemed to be for different issues, from whether AWS is a good fit for our subject matter, dislike of AWS as a whole, and in one case, the dislike of SE's corporate parent. There was also a downvoted (-4) answer (now deleted) asking whether having the sponsor's logo on our site was necessary.
I would assume that SE is going to vet their sponsors well (So CMOT Dibbler's house of canned cat meat isn't going to sponsor pets or cooking). I haven't seen a case of a broad strenuous objection to a sponsorship, either on my site, or others so far.