I can't find one and it might be something you could start if you believe in the cause. However, I don't see it as being a feasible option, in the way you probably mean it. There are currently so many animals without homes that they can't keep them in the shelters they currently have. They have to destroy many, many animals that could have a good home life, otherwise. They simply don't have the space or resources to care for more than they are already caring for. So it's unlikely that most people will help a pet that already has a home. They might not be getting the best home, but they have it better than the ones who are put down.
Another issue you have is dishonest benefactees. Just like now, you would have so many people abusing programs designed to help people, like food stamps and medicade. I was once working at a temp job and overheard the supervisor telling someone that her food stamps were coming in and she'd be ready to sell them. That program was meant to help people who couldn't feed their families and there are people who can obviously afford food, because they're selling their food stamps to buy extra unapproved items. That's money that could have gone to someone who really needed it. You'll have the same thing. You'll have people that can afford to take good care of their pets, but will refuse to, because they have a work around they can exploit in the charity. The current system is better, because the dogs that are helped are in the sole custody of the care providers and can then be rehomed to someone who will foot the bill for future care.
If someone were to create a system to help pets before they're rehomed, that would be great, but you'd need to really police it and think about how people could screw over the system. For instance, it wouldn't be hard at all to design a website where you can add all kinds of information. You have lots of people who are moving into areas that don't allow pets. You could have a place where citizens can submit the names of dwellings that would allow them to keep their pets. If people have medical bills for their pet and they can't afford to pay for them, so are looking at putting their pet down, you can potentially find an institution willing to help them until they can pay it back, though most vets will work with you and get you on a payment plan. Providing links to volunteer trainers that have some kind of basic certification and are willing to help people with their behavior problems, which cause many dogs to be rehomed. Several of these ideas can only benefit the pet without the owner being able to scam the charity.
Something I think is very important that such an organization could do is fund groups going into schools to talk to children about proper pet care. Giving children an idea is a powerful thing. I see some groups who realize it and some who don't. Marketers certainly have a grasp on how giving a child and idea can have a huge impact. I can give you examples. Happy Meal toys are an excellent example. This included toys, collectable glasses, etc... The fast food chains can put one commercial on tv and show one of the toys with an exciting commercial and the kids really want it. Even if the adult knows it's a piece of junk and an overpriced, unhealthy meal, they will often get it because they want their child to be happy. Once there, they see there are often a whole set to collect and you can't request one, you just have to keep buying them till you get them all. So where a parent might only buy a single happy meal every few weeks to months, now they get into a loop of buying many in a small period of time, massively driving up sales.
Another example from my own childhood. In kindergarten I was taught to put a line over the capital 'J' in my name. Then in first grade they said some educators had changed their minds and we shouldn't do it anymore. Well, the first way I'd learned to do it was with a line over the top and it was such a strong impulse that I was unwilling to change it. The next year they came back and said they were back to putting lines over the J's. I'd been doing it all along. I was also taught in school that Pluto was a planet. I still consider it to be one and I heard recently that they might be adding it back in.
The point to those stories is that kids get their first impression on many subjects in school and they stick with them. There are plenty of things my parents do one way, but because I was taught in school that they were done another way, I do them that way. If it's something major I might not make a big deal about it with them, but I just do things the way I think they should be done.
That same concept will work with kids now. If people go into schools they can give kids the first impressions they should have. You can help nip fear of animals in the bud, you can explain why spaying and neutering are important, you can tell them that buying a puppy from a breeder is perfectly fine, but they need to find a responsible breeder, also that their are many great dogs in shelters that need good homes.
I know at least two adults right now who have un-neutered male dogs. One of them lets his wander. I can ask them both why they don't fix them and they'll both say that it's because they don't want to take away their manhood or make them feel like less of a man. The truth is that they're projecting and that as opposed to making them feel less, you're reducing frustration, testosterone, behavioral issues, and potential medical issues. If you stick it in their heads as kids, they'll lean that way in the future.
For those people reading my remark about buying bred puppies and shaking their heads, that's okay. I still think buying a puppy is just fine. I heard the same arguments when the horse market fell through. They'd try to castrate you if you mentioned buying instead of rescuing. While I really support rescue animals and have had many over the years, I've also had good bred dogs, bought from responsible breeders. Dog breeds have been bred for many, many, many years. They were bred for specific purposes and there is nothing wrong with supporting and celebrating those breeds any more than there is in rescuing a shelter animal. The charity could also offer information on good breeder to contact in your area as well. That would be an excellent resource. Also, what to look for in a good breeder. There is lots of information out there, but having it consolidated in one place really makes a difference. This site is a good example.
I've been saying kids all along, but adults will benefit from all the educational resources as well. The more good information that is out there the better. So I know I've gotten kind of long winded with this reply, but I feel like it's all on topic with your question. While I couldn't find any specific charities in my searches that do what you are asking about, I do think that you could start or encourage an organization to develop that could provide a large number of resources to help owners and give them the tools and resources that if they were lacking, could cause an owner to loose their beloved pet. Crowd sourcing is an excellent device and if you can find a forum or a place suggested by this site, where you can find like minded people who think this is a good idea. Once you get a group of people together on something like this, it can really take off. Maybe you find someone that either has some extra storage room on a server or they know a business that would be willing donate money to pay the minor fee to get the site storage space and the domain name(s). Maybe there is a web developer who loves his cat and would be willing to donate his time to develop the site and do the coding. Maybe he or she will help maintain it or other volunteers will, though a good developer will make a back end that is easy for users to maintain the site. The point is it can easily snowball and before you know it you have an organization and all kinds of people looking to help out. Good luck and I hope I answered your question.