That's definitely one to discuss with your mental healthcare provider. If a person is emotionally dependent on the support of an emotional support animal, not only does that support mechanism suddenly cease, they also have to deal with grieving for the lost friend.
The absolute safest course of action, of course, is not to put all your eggs in one basket. It is important to have several different support systems in place. There are medical providers, family, pets, and friends (both Internet and real-life). One of the things I've noticed is that lots of people with disabilities, and particularly those with mental illness disabilities, is a tendency to shut in so that their primary contact with the outside world is through the internet. They have few real-life friends. This is dangerous!
So make sure you have a well planned support system that includes several different types and several different resources. That way, if one resource is unavailable (perhaps you can't get ahold of them when needed), you still have other resources to turn to.
When you know an ESA is nearing the end of his or her life, I think it is reasonable for the doctor to prescribe a second animal, who will eventually take the place of the older one. For example, I have my service dog who is 11. The average life expectancy of his breed is 10.3 years. If he were an emotional support animal, it might be reasonable for me to ask my landlord to permit me to acquire a second dog, given my current dogs advanced age, so that I wouldn't be left alone during the grieving process.
The whole thing is about reasonableness. I think the scenario above would be a reasonable accommodation, but you'd need to convince a landlord that it is also. I have succeeded in negotiating something along these lines for one of my clients and her landlord was really quite willing based on the doctor's letter we submitted. They too saw the logic involved.
I do agree with your sister that now is the time to plan ahead. At the time you lose something like an ESA is the absolute worst time to be dealing with this kind of issue as well.
