I think your question is ultimately about trust, which is a soft factor that kind of enables us to work together here.
Your own recent experience highlighted to you the risk that internet services might disappear. And that's reasonable, because they do. In this case the risk is very, very low though over the next years. The core of the public Q&A is engineers (the number of programmers is really surprisingly small), community moderators and maybe further support staff and the infrastructure to run the servers. If the traffic that Stack Overflow generates (billions of visits every year) continues, the generated amount of advertisement revenue is likely more than enough to keep the lights on for a very long time (making lots of profit is a different thing though, I guess that selling the software that powers the platform is a better approach for that).
With this risk in mind, what you basically wanted in this question is a last will, maybe even some kind of contractual obligation or trusted third party as guardian of the data. And there is such a thing as part of the quarterly data dumps stored on the Internet Archive. It also helps that the data is under an open and quite compatible license (and now also with clarity about the version of the license).
When Jeff Atwood announced the data dumps in June 2009, he kind of created the trust that whatever may happen (except when the Internet Archive goes down and all other stored recent copies of the data dump too), the content will be there and will be accessible. Maybe this was a contributing factor to the success that followed.
To directly answer your question: No, there is no official succession or continuation plan published that I'm aware of and I have read many meta questions over the years and have just searched for terms like "what happens if Stack Overflow went down / shut down / goes bankrupt / goes out of business" and I only found a couple of Quora questions about it. Indeed there is no real advantage for the company to write down a legally binding last will, except maybe for generating even more trust. And they don't need to, because the data dumps and the license already generate significantly more trust than the user content stored with Facebook for example (a last will for Facebook might actually be a really good idea).
Likewise I have not heard of any substantial plans to convert the Stack Exchange sites into a charitable organization. The company still wants to make profit with these platforms and the other products/services. The investors who just recently put new money in the company surely expect an adequate return.
To summarize: As longs as the quarterly data dumps keep coming and are intact (how to check that might actually be a very interesting question), don't worry - all will be well. If they stop, think twice.
This is just as a bit of extended commentary to the answer of Journeyman Geek.