I'm able to derive the Schwarzschild solution under the assumptions that the metric is (1) static (2) spherically symmetric and that the space is the vacuum. However, I have read that the Schwarzschild solution can be found assuming only that the metric is a spherically symmetric vacuum. How would the Schwarzschild solution be derived under these weaker conditions?
There is a theorem which states that any spherically symmetric solution to the vacuum equations is also necessarily static and asymptotically flat. It is known as Birkhoff's theorem. Chapter 4 of Straumann (2013) contains a full proof.