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We will only have a full medical test and assessment on Carvajal in next several days but it's not like he will recover after a few days. It's not a minor injury.
According to
La Liga Injuries & Suspensions from Sportsgambler, this injury will take Carvajal about 2 months and he will be back in late of November.
He is very professional and already played many seasons for Real Madrid. He is important for them in defense, attacking support and he is even able to score important goals for Real Madrid. This is a smart, skillful and professional player and Real Madrid as well as Carlo Ancelotti will miss this player in next several weeks.
Carvajal has torn his ACL. It's a major injury, and it takes months to recover.
About Real Madrid, they are now facing serious issues in their defensive line. They lost lots of key players such as Courtois, Alaba, and now Carvajal. They have Vazquez, who can fill Carvajal's position. But, I'm doubting him to play at the same level as Carvajal did.
They obviously need to add more players to their squad. A very busy schedule, and intensive games are ruining their players.
Madrid managed to win the match against Villareal with a convincing score, the defeat to Lille in the previous match made them motivated and when the match just started, they immediately attacked the Villareal defense, so the score of 2-0 for Madrid victory finally made them equal the points from Barcelona who have not played their 9th match today, the victory actually had to be paid dearly by Madrid over villareal after Dani Carvajal and Vinicius suffered injuries in the match, And with their current injuries make many of Madrid key players are injured at the same time, so obviously this will make their defence very vulnerable.
Lucas Vázquez could indeed be a temporary replacement, but as you said, his quality may not be on par with Carvajal, because I think he is a pretty solid defender and often helps Madrid attacking line so far, although maybe Madrid have a capable squad depth but the number of key players who are injured will of course gradually affect their performance this season.
