Pretty much all of the Catholic Church's money comes from passing the hat in one form of another, and its absolutely normal for a new initiative by a Catholic institution to be accompanied by an effort to seek direct donations to support that particular initiative.
For the same reason funded startup's use crowd funding. It demonstrates that this is something a lot of people care about and builds community buy-in around a cause. The people that run this digitization program at the vatican have bosses too and they need to justify that their project matters and deserves more time and attention.
How do you think they got that way? Institutionalized panhandling.
In the same way that you might donate to a soup kitchen run by a Church fellowship, because the money goes to feeding hungry people, you might donate to manuscript preservation.
That people in the Church donate their time and money towards such causes and ask for other people to donate their money doesn't seem wrong.
They talk about "saving" the documents. Are they going to just let them rot away if we don't cough up?