How can I add a delay to an integrated laptop microphone?

Unfortunately, it doesn't appear that you will be able to fix your problem. It sounds like you will probably use the same setup that your friends do to let them keep up. On the other hand, there might be a piece of software on the internet that allows the delay, but you'd have to dig for it... I honestly can't think of another way for you to do it... I hope you figure it out!