Solving that problem isn't as hard as you might think.
Minimal Viable Team
Start by looking at the deliverables due over the next 90 days and figure out what the minimally viable team looks like that can keep the projects on schedule. We're not talking about beating the deadline or out-performing expectations, we're simply talking about who are the mission-critical individuals without whom the project literally stops. Usually, these are identified individuals, but in some cases, it can be a role within a competency (which anyone within that competency in that role can fill).
If members of the MVT have requests in for vacation time, then the question becomes:
a) Is there someone else on their team who can step into their role as MVT member?
b) If not, is there any way they can manage their work on the project remotely for the period they'll be out?
Understandably, most people don't want to go on vacation knowing that they're going to wind up working during their time "off", but the realities of today are what they are and such a requirement is not uncommon. Best practice is to provide some sort of compromise in these circumstances - such as extending their vacation leave in exchange for their availability during the crucial time period.
Faced with the option of not getting their vacation request approved or having to be available during their time away, most employees accept the availability clause and are grateful for the chance to get out of the office.
Client Requests
Another common way of handling the holiday time crunch is just talking to the client about how critical it really is for every deliverable to be completed by end-of-month or end-of-quarter. Sometimes this can result in a postponement of some of the work into the following quarter, which frees up agency resources.
Outsourcing
Outsourcing is the least favorite option because it decimates the agency's profit, but when you're faced with the specter of employees willing to quit if they can't get their vacation time approved and you simply can't fill an MVT, it's better to outsource the work and keep the client than drop the ball and lose the account.
In some cases, you can bring in freelance talent to replace the MVT role(s) necessary, which minimizes the impact to the agency's profit, but last-minute freelance talent is expensive (especially during the holidays) and in many cases, the complexity of deliverables prevents this strategy from being viable (meaning that the amount of time it would take a freelancer to get up to speed on the project would be prohibitive to successful completion of the deliverable).
Consolidation Prevents Chaos
One thing is for sure - when an employee goes through the proper channels to request time off and doesn't hear back about whether that time has been approved or not - they're going to assume it's been approved. Count on it. They buy their airline tickets, they tell their family they're coming and they come unglued when you tell them they can't go because you didn't realize they requested the time off - it wasn't reflected in the timesheet system or the Project Management software.