Who believes the incarnation and resurrection wouldn't have happened without Mary's faith? A group called the Steering Committee for the National Consecration of Ireland to the Immaculate Heart of Mary is, as the name suggests, seeking the consecration of the nation of Ireland to the Immaculate Heart of Mary. In their FAQ, they say (in the answer to the last question):


  Some theologians teach that without Mary’s free fiat of “Let it be done to me according to thy word” God would not have worked the incarnation and others go as far as to say that without her perfect faith at the foot of the cross He would not have worked the resurrection.


Who are these theologians, and how generally accepted is their teaching in the Catholic Church? I'm curious whether there are important Catholic theologians who disagree - basically, whether there's a debate about it.

For the second part specifically - do these theologians suggest that in this case, Jesus would have simply died and not been resurrected? Do they believe that Jesus would have died, and yet our redemption would not have come about?

(I thought we had a past question which touched on the Annunciation part, but I can't find it right now. In any case, I'm pretty sure the second part is new.)
With or without Mary, God does what he intends to do. God tells us so at Isaiah 55:11


  "So my word that goes out of my mouth will be.
  It will not return to me without results,
  But it will certainly accomplish whatever is my delight,
  And it will have sure success in what I send it to do."


If Mary had not found favor in gods eyes, she would not have been used for his purpose. God could have picked any women that fit the prophecy. He chose her because of the qualities he found in her.

Regardless of Mary, Jesus did all that his father instructed him to do. We see from further accounts from the bible, that Jesus's purpose was more than the ransom sacrifice. Jesus also instructed us in what we need to do in order to be saved. Jesus of course met all those requirements and was resurrected. 

As for the theologians that that website's FAQ is talking about, I cant find any good research on it. that website has absolutely nothing to help me research, besides key terms that I'm looking for on my own.

I found a few articles, but I'm not sure they will answer your question.

Devotion to the sacred Heart of Jesus, [deleted this one as it just reiterates what you said].... ,Beside the Cross of Jesus  &lt;- this site here has the teachings of the catholic church. 

It looks like they are inferring that without Mary, Jesus wouldn't have been born in the flesh. I cant find any article that exclusively says that without Mary's faith Jesus would have not been resurrected. 

Also I find it very frustrating that none of these sites show where they got all this information, so I can research it.  Vatican is referenced for a few on that site. 
CatholicEducation gives some more of the same insight.