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We do have our own system that works for us, where labels are our main attributes for almost everything (employees name, type of leave - sick day, holiday, annual leave, wedding, approval, team), together with that you would also have a total number of days a person can take off in a year (which is a number attribute, first item in the screenshot below), then date attributes (to know exactly when those days were taken), we also have a team label (support, dev, marketing, growth, and so much more) - which will allow you to create tabs for each individual team and track days off for each team in a separate view/tab.
Here is a screenshot of how that looks for me (example):
If your employees have access to this board then they can simply come in - create an additional item, label it appropriately for their type of leave, set number of days and be done with it.
Or you can create a request a vacation/holiday/days off form, and let them fill it out whenever they need to take a few days off, which you will later on approve their requests and inform your employees about it.
(forms do help because you can get notifications whenever someone fills out the vacation form)
Hopefully this helps at least a little bit, also - feel free to let us know what you think about it.