Some members do other things as managing a club, train tactics, read an article, etc. And that doesn't mean they have time or desire to also play daily (I mean playing it seriously, not just move the pieces) at that moment.
If smb abuse, report it!
Yes, 100% right! 
That system works with a small number of games, but try it for +100 games, as is fairly common online, and you will see the logistical problem of keeping all that vacation time straight.
There are enough people playing over 100 games that the system needs to be workable for them too.
Tracking vacation time separately per game would be somewhat awkward for even 25-30 games.
If we assume a player doesn't want to lose any game on time, they just need to know how much vacation time is remaining in the game with the least amount of vacation time.
Five days in a row is not sufficient. It's very common for someone to legitimately take a week or two off.
And having no limit to the number of five-day vacations would essentially allow someone to add five days to every daily time control, turning 3-day games into 8-day games. This would allow people to drag out lost games much longer.
Yes, hmmm.... There also needs to be a way where you can report if a certain player is abusing the vacation rules.
You can use the report button on his page just like for any other infraction.
But the vacation system should be set up in a way that's ideal for legitimate uses, and, ideally, doesn't cater so much to abuse in that manner. Some cases of abuse can be prevented by simply changing the way the system works to make the abuse impossible.
Why is anything gained by tracking vacation time on a per-game basis rather than overall? Doesn't that just complicate things, especially for people playing a large number of games?
I don't think there's any need for significant changes, other than to decrease the maximum total time. If you can never have more than, say, a month for diamond members, that already significantly limits how much you can stall.
The current system is described here: https://support.chess.com/customer/en/portal/articles/1444787-how-does-vacation-work-how-much-time-do-i-get-?b_id=12321
You get an initial 7 days. Twice a month, on the 1st and 15th, you get an additional 1 to 2.5 days added, depending on membership level. This caps out at between 30 and 90 days, depending on membership level.
I don't see any reason to change anything other than to reduce the cap.
Yes, I see what you are saying, but the reason that I like my method is that if players who actually need lots of vacation time, can get it, while players who abuse vacation cannot just sit on the game for weeks without playing.
If they abuse vacation they can be reported and staff can go into their accounts to resign the games?
I don’t think that the staff can log into their accounts, but they can lock the player out, making them lose on time
Why is anything gained by tracking vacation time on a per-game basis rather than overall? Doesn't that just complicate things, especially for people playing a large number of games?
I don't think there's any need for significant changes, other than to decrease the maximum total time. If you can never have more than, say, a month for diamond members, that already significantly limits how much you can stall.
The current system is described here: https://support.chess.com/customer/en/portal/articles/1444787-how-does-vacation-work-how-much-time-do-i-get-?b_id=12321
You get an initial 7 days. Twice a month, on the 1st and 15th, you get an additional 1 to 2.5 days added, depending on membership level. This caps out at between 30 and 90 days, depending on membership level.
I don't see any reason to change anything other than to reduce the cap.
Also make them take it in larger chunks. This is a little too much, but imagine 4 weeks of vacation max, and the minimum usage is 1 week chunks.
So a person can go on vacation 4 times a year, max.
I don’t think that the staff can log into their accounts, but they can lock the player out, making them lose on time
I think staff can. Have you not seen banned people are still green here sometimes?
Yes, I see what you are saying, but the reason that I like my method is that if players who actually need lots of vacation time, can get it, while players who abuse vacation cannot just sit on the game for weeks without playing.
Doesn't my suggestion of simply reducing the maximum accomplish the same thing? Sure, you can sit for maybe 3-4 weeks, but then you're completely out of vacation time and it will take months to recharge.
Some members do other things as managing a club, train tactics, read an article, etc. And that doesn't mean they have time or desire to also play daily (I mean playing it seriously, not just move the pieces) at that moment.
If smb abuse, report it!
Agree.