Would a Schwarzschild black hole “appear” to be a sphere in all reference frames?

I am wondering if all observers would agree that a non-rotating, non-charged black hole is spherical (i.e. there is no reference frame where one would measure it to be oblong in one dimension). I assume that this is true, but with relativity I have learned not to assume.

As long as the observer is stationary, then yes. The Schwarzschild metric has complete spherical symmetry: in fancy terms, you can say that given any two timelike trajectories at fixed values of $$(r, \theta, \varphi)$$ and at the same $$r$$, there is always an isometry taking one to the other. You can also calculate the shape of the black hole's shadow: it turns out to be a circle, a bit larger than the black hole itself, and independent of the observer's position.