devtrackers.gg
According to everything I've read online (articles, blog posts, etc.), the Vault opens after 900 progress points, where each fifth common gives 1 point and each fifth uncommon gives 3 points. This is confirmed by using the Internet Archive Wayback Machine to look at a snapshot of the official Wizards website on rarity and drop rates from June 19, 2019:
https://web.archive.org/web/20190619221143/https://magic.wizards.com/en/promotions/drop-rates
However, according to the live version of that same website, the vault now requires 1000 progress points to open:
https://magic.wizards.com/en/promotions/drop-rates
External link →