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They have a department responsible for this, among other things.

https://about.vanguard.com/investment-stewardship/perspectiv...

I only opened up the "500 Index Fund" PDF but it was interesting to note that they voted against almost every proposal that was brought by shareholders

Yes, and they voted against the proposals mentioned in the article, except for equal share class voting rights.

They probably vote in the best interests of other people. They possibly even sell the votes... (Is that illegal?)

Yes, it is.