I think you can still make a game of this, but the campaign has just shifted into an example of asymmetrical warfare with the runners on one side and the Mob family in question on the other.
First off, there is no way I can see that the mob boss can let them walk away from this; he or she will lose too much face and prestige, and the competitors out there (the other Mafia families, the yakuza, the Triads, etc.) will start moving in if they perceive weakness.
Come to think of it, which mob did they piss off? The precise nature of the response will differ a bit - for example, a yakuza oyuban or Mafia don may pull in reinforcements from their (very large) parent and affiliated organizations. Vory gangsters will likely opt for a more direct and personal approach, probably something involving vice grips, blow torches and the character's reproductive organs.
The thing about big criminal organizations is that they will also have allies - and not all of them will be wise guys and street enforcers. Cops and members of the DA's office will be on the payroll. They'll have their hooks into various corporations at differing levels, and may actually own some smaller local corps outright. Higher level politicos might owe something to the capo as well.
So the characters should expect to see their names posted up by the local law enforcement as "terrorists" wanted in conjunction with all those bombings. Their fake IDs will start getting burned just as word goes out among the forger and fake ID vendors to cut them off, which at the very least will force them to more expensive to maintain fake identities. Known associates (high loyalty contacts, dependents, etc.) will have the heat turned up on them - perhaps not killed outright, but at the least discouraged from working with the characters. Bounties will be placed on their heads, both open and among the underground.
Oh, and the mob boss can hire shadowrunners of his or her own to supplement these forces. And don't forget paracritters....
Now, this doesn't mean the end of the game. The characters can try turning to other criminal organizations for protection, or an alliance. This could result in a tense war between the criminal organizations that eventually pulls in law enforcement, but might succeed in hitting a point where "mutually assured destruction" becomes less desirable to the mob than cutting a deal.
You could also find ways of dangling the alternative of escape to some other locale, possibly overseas, which likely won't stop the lethal response of the mobsters, but would likely attenuate it somewhat. This could be coupled with the alliance above - for example, if you're fleeing the Mafia, cutting a deal with the Yakuza in exchange for safe passage to Neo-Tokyo might be worthwhile.
Or they could cut a deal with a dragon.
Either way, it's going to get bloody, and it would not be excessive for one or more PCs to go down. They poked the bear with a stick, and the bear is REALLY angry.
Edited to add: the points up-thread about talking with the players, and explaining that the D&D/heroic fantasy tropes don't apply, is excellent advice.