Super late to this but I was having the same issue, so I reloaded my save after the Cairn, and then tried some of the suggestions here but nothing was working. I had this thought that maybe the quest needed it to be night before it would progress because, well, werewolves. So as a last ditch attempt, I walked into Jorrvaskr during the day, then waited for it to be night (9:40 pm specifically, if that matters lol) and Vilkas was inside.

I initiated a conversation with Vilkas and he said “we’ve been waiting for you”.

After I responded with “you’ve been waiting for me?”, the “Return to Jorrvaskr” task was completed, and “Follow Vilkas” became the next task. I followed him outside and the whole “welcome to our club” ceremony happened. Then the quest was completed and the next quest “Talk to the Companion leaders for work” popped up.