Taking notes is a lot more important than people think. I would encourage you/them to definitely have someone in charge of jotting down little notes during rehearsals, even if it's not a formal report format.
Also, I just did my first show without a formal prompt book, and I was totally lost. I felt incredibly disorganized and I will never not have one again. Having all the show information in one place is extremely helpful, especially if you can't always be there. Schedules, medical information, the script, notes... keeping it all together just helps the show run so much more smoothly!
Half of the things I have picked up as a stage manager have been things other stage managers have used and I've observed! I would just start doing things (keeping notes, assembling the book, encourage daily communications, that kind of stuff!). Sometimes it takes one person being really into it to gently nudge others into trying it.