[Hobbes]

Unofficial announcement here, then:

I was surprised that it was a complete sweep, but bottom line, by canon, the three Mita-Gumi contacts are deceased.

It will be more clearly detailed in the next FAQ release, and we will find a place for it in the Season 10 Missions themselves.

Will there be some sort of work around for the teams that opted to side with the Mita-Gumi?

Or are they just fated to always be looking over their shoulders when they get calls/missions from/for the Inagawa-kai?

Welcome to a shared campaign!  YMMV.  There are some missions where player actions have an impact on other missions.  If it doesn't match with what the PC did, they just have to head cannon/hand waive.  I've played the same missions with multiple characters and had different outcomes, played with different players who did previous missions differently... you just kinda have to roll with it sometimes. 

If a GM has a table full of PCs that all made similar choices (or played the same game even) a Mission GM can improv a bit to keep the story line intact. 

[Stainless Steel Devil Rat]

SRM FAQ v1.3 pg 30:

I’m playing a Mission and there appears to be a narrative paradox, I have a Contact from a previous Mission
but the other players are telling me that he was killed in their last Mission. Do I still have the Contact?
Sometimes, there will be instances when characters have experienced very different versions of the same
events or some will have participated in adventures that others haven’t and the disposition of particular contacts will
be different. When this occurs, the player who possesses the contact still has access to that contact and the player for
whom that contact no longer exists should play it off as if the contact being used is different from the one they know
to be dead. A little bit off hand-waving to move things along.

[prophet42]

I suppose it'll just be a weird situation then for my team.

Perhaps the Yakuza (in general) & the Inagawa-kai (in particular) won't be the driving force for most of the missions in Season 10.

[Jayde Moon]

The focus does change in Season 10.

GMs and players are encouraged to alter the stories just enough to make it fit.

Went after and killed Yoriko in 09-06?  Alter the story so that it was another Inagawa-kai street boss, not Yoriko in particular.  Now, she's still alive and doesn't have a memory of you trying to gut her.

As Hobbes put it, welcome to the shared campaign.