Great question — answering as the UphillTech team:
TL;DR: the anime studio doesn't own the complete rights to the anime, so in most cases there will be no problem in the case the studio goes under. Long version: The owner of the rights to an anime in modern times is almost always a 制作委員会, which I usually translate as "production partnership". This is a partnership that is set up in order to fund the anime, while splitting the risk with other sponsors. This includes companies like the publisher of the original work, the music company, figure companies, advertising companies, TV stations, etc. This may or may not include the anime production studio itself. Either way, in the case the studio ceases to exist, this would not affect any agreement...
Full answer by Jimmy: https://anime.stackexchange.com/a/43349 (CC BY-SA 4.0).
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