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Mozilla employs a small number of people to work on Rust fulltime, and many of the Servo people contributed to Rust too, even if it wasn't their job. They also pay for the hosting bill for crates.io. They also own a trademark on Rust, Cargo, and the logos of both. Two people from the Rust team have posted about their situation, one was laid off and one was not. Unsure about the others. Many of the Servo folks (and possibly all, it's not 100% clear yet but it doesn't look good) have been laid off.
The vast majority of contributors and team members are non-Mozilla, though the Mozilla folk are very important. Rust will survive. This situation is very painful, and it has the possibility of being more so, but (thanks to a lot of work from people (not speaking of myself here) who used to be at Mozilla!) Rust is bigger than Mozilla, and has been on a long road towards diversification. Some other companies have stepped up to pay a bunch of the bills that Mozilla used to pay, namely Amazon and Microsoft. Here's hoping that we get even more involvement in the future.