This sounds like an edge-case, and the solution seems rather simple: implement a policy so that only X% of employees in any department can be on vacation at any given time. Just like employees in retail have to coordinate their shifts (and have to find substitutes if they can't cover their shift), you can have this policy to ensure business continuity.
That helps, but doesn't completely solve the problem. This is actually done by many companies in Germany that I know of.
However, if you set the X% relatively high (e.g. 40%) then you still have the problem that the whole department might not function that well any more during this time.
If you set it very low on the other hand (e.g. 10%) then many people won't be able to take their vacation during the summer months and they will be pissed.
Sure, there's a way to find a middle ground that works out somehow, but still the original problem isn't completely solved.
