I have a few vacation days saved up that will expire soon. However, this is the busiest time of year at my office. How do I go about using these days without putting an undue burden on my coworkers?
In a company that I worked for previously where I had a similar issue, by talking to our managers, we could get the expiration date on the vacation days extended by a few months. This was as a win/win for both employee and employer: the employee doesn't lose his/her days of vacation, and the employer doesn't lose an employee during an already stressful time of year. It might be worth it to see if your management is willing to go this route.
In my workplace, the answer is that if management really needs you not to take vacation they will make sure you don't lose the days -- they'll issue you Management Directed Time Off to make up for it. In fact I'm doing exactly that, unofficially carrying five days over that would otherwise evaporate.
In some organizations, you need approval to use vacation time. Your manager is the one who can approve your time off. In other organizations, you don't need formal approval, but it is usually expected that you will coordinate with your leads, team, and/or coworkers to ensure that there is necessary coverage on your proposed vacation days. If you have days off in mind, you can simply talk to your team leader or your coworkers to see if there are any problems. Since it is a busy time of the year, you should be prepared to not be able to use all of the time that you have available.
If you're serious about getting those vacation leave credits in that time of the year (ex. hospitalization), aside from having the talk with the manager, you should also tell your colleagues about the tasks you are about to leave beforehand that needed to be done before you come back. If any of those tasks are urgent and they had to do it in your stead, you should give them some documentation, some processes to follow. With a demonstration if needed.
Or if you are not really keen on having a vacation, then you should ask your manager to extend the expiration date of your vacation days. @R_Kapp mentioned about extending the expiration dates if leave credits and its merits. I don't presume if this applies to your company, but the answer is always no if you don't ask them to.