If everyone is showing in friends and family list on each other's accounts in MDE, then everyone can try using their own account.
However, if EVERYONE doesn't show up in everyone else's MDE, you could end up getting a different BG for some people if trying from multiple accounts.
Here's what I mean... I have an account that has my wife, son, etc. on it. They don't have their own accounts. They just exist as avatars/profiles that I created on my account and I manage everyone's plans.
I'm linked to my brother's account. We were at DHS with his family in the fall. He was trying for a BG on his phone (using his account) and I was on my phone (using my account).
He was a little faster getting a BG than I was and I was included in his BG, but not my other family members who don't have their own accounts. I was successful getting a BG for the other people on my account, and we were able to get everyone combined into the same BG by visiting the blue umbrella team, so it all worked out.
I assume if I'd gotten through first, it would have included everyone in the BG because I can see all of his family members (and was even able to make park reservations recently for them from my account). I'll have to verify with him, but I think it's because he actually created accounts for all of his family members that are linked together.
Apparently, only
I am linked to his account, but the other family member profiles I created within my account are not linked to his.
My point is, if you have multiple people trying for a BG from multiple accounts, it's possible that you end up with some people getting a different BG (or worse, they get "orphaned" if the only person whose account can "see" them is not successful getting a BG).
You could probably check ahead of time with everyone trying to make sure the entire party shows up in everyone's accounts. If they can see everyone, then whoever gets in first should get a BG for the entire group. But if some members of your party are not visible in everyone's accounts, they might not be included in the BG if the person who gets in can't "see" them in MDE.