Can my friend remarry civilly before her fiance’s annulment, then have the marriage blessed?

No, this is definitely not okay. A Catholic cannot validly marry outside the Church without permission (cf. Code of Canon Law 1108), so a civil ceremony and subsequent Church blessing is not a moral option. The Church presumes the first marriage to be valid unless and until proven otherwise through the annulment process, so your friend should not even be dating him.

Consider this scenario: The couple marries civilly and then the Church’s marriage tribunal finds his first marriage to be valid—he doesn’t get the annulment. Your friend will be civilly married to a man whom the Church recognizes is already married to someone else. She’ll actually be living and sleeping with another woman’s husband. She won’t be able to have her marriage blessed in the Church. And, as long as she continues to live as though she’s married to this man, she’ll be barred from receiving the Eucharist and won’t be able to receive absolution through the sacrament of penance. What a mess!

Your friend needs to take a step back and wait for the results of the annulment process. Only then will she know whether this relationship is something she can morally pursue.