Why can't I save on a hotel or camp site? - Setting a 'Home Base'
In your main neighbourhoods you would save by returning to your home residential lot, or by saving while on a business lot (if you have the Open For Business pack installed); in the vacation destinations the game first needs to know where the temporary home lot is going to be, you do this by setting a 'home base'. Until you set a 'home base' you will be unable to access the save function.
To set an owned lot as 'home base'
If your sims own their own holiday home in that neighbourhood and you want them to stay there, just move them straight into the lot, this sets this lot as the 'home base' automatically and enables the save function. You can now save and have your sims visit any of the other lots.
To set a campground as 'home base'
If your sims will be staying at a camping ground, just arriving at the lot is not sufficient to set the lot as 'home base' and you are still unable to save. This is because you have the option to change your mind and take them elsewhere instead.
In order to save on this lot you must first have one of your playable sims click on any of the available tents and use the 'Check In' interaction. Once you have done this it sets the lot as 'home base' and the save function is enabled.
To set a hotel as 'home base'
If your sims will be staying at a hotel, just arriving at the lot is not sufficient to set the lot as 'home base' and you are still unable to save. This is because you have the option to change your mind and take them elsewhere instead.
In order to save save on this lot you must first have one of your playable sims reserve one or more suites by clicking on the suite door and using the 'Reserve Suite' interaction, you must then have one of your playable sims check in to the hotel by clicking on the front desk and choosing the 'Check In' interaction. Once you have done this it sets the lot as 'home base' and the save function is enabled.
Once you have set the 'home base' for your sims you can then save and take them to any other lots in the vacation neighbourhood, including other hotels and camp sites, however you must always return to the 'home base' to save.
Should your sims check out of their hotel or camp site at any point, the save function disables again until you either check back in, go to another lot and check in, arrive at an owned holiday lot or return home to the base neighbourhood.
Sims staying in their owned holiday homes cannot set a different 'home base' during their current vacation, to change their 'home base' to a different lot they must first travel back home and book and pay for another vacation.