There is no RAW on this topic.
I have not reviewed every 5e book, but I have several of them, and to the best of my knowledge, this is never addressed in any of them. More than that, I don't expect it to be addressed in any of them in any detail because this is simply not the type of thing that tends to get described in a lot of detail in 5e.
The fact that characters can die of old age is mentioned many times, and the average life-spans of many fantasy races are mentioned, but the details of aging and thus an answer to this question is not something likely to be covered in much detail in books about high adventure.
(Magical aging is a separate thing and is discussed multiple times. But in general the specifics when magical aging are involved are usually discussed alongside the effects).
When appropriate for the story, taking into account the species.
I have a hard time coming up with a scenario where you would want to roll for this. Instead, if it becomes relevant, I recommend letting the story dictate it. If it is dramatically appropriate in story for a character to die of old age, and they are not so young that it would be ludicrous, then they do (Think Yoda in ROTJ. The timing was very convenient story wise.) If the story would be served by them living, and they are not ludicrously old for their species, then they do not die of old age (and if they are ludicrously old, but their death would be inconvenient, there are various magics that could be invoked to buy them a little more time while maintaining verisimilitude).
If you really want to use dice, then consider using a percentage and basing it off of actuarial tables.
If you really want to use dice for this, then at least for humans you can look to real world actuarial tables like this one.
There is a death probability chart for each age. You can roll a percentage (use a d100 or 2d10 with one of them representing the tens place and the other the ones place). If they beat the odds, they won't die of old age for another year. If they don't beat the odds, they will die at some point in that year.
This of course is not perfect because most actuarial tables look at death from any cause rather than just old age (It can in fact be difficult in real life to say whether someone died of "old age" or not. If someone died from an identifiable disease that a younger person probably would have survived, did they die from old age?) Still, it makes a decent starting point if you want to roll.
You may not want to use a constitution modifier at all. For one thing, 5e at least to the best of my knowledge does not have rules for reducing constitution with age, which is an expected result of aging in the real world (though how much gets complicated and depends on how well someone has been taking care of themselves, genetics, environmental factors, etc). I would simply disregarding constitution. But if you want to factor it in while using the actuarial tables, consider adding the entire constitution number to the percentage roll. This could result in some people living exceptionally long lives, but that might be expected at least of people like adventurers that are meant to be exceptional.