Is it possible to convert Option<Result<T, E>> to a Result<Option<T>, E> without using match?

In Rust 1.33, transpose() is stable, so you can just call it:

fn main() {
    let x: Result<Option<i32>, ()> = Ok(Some(5));
    let y: Option<Result<i32, ()>> = Some(Ok(5));
    assert_eq!(x, y.transpose());
}

You can use Option::map_or():

val.map_or(Ok(None), |v| v.map(Some))