How can a flying wing style airplane recover from a spin without a vertical stabilizer?
The answer depends on the type of flying wing aircraft.

Drag rudders

The B2 for example is an example of a flying wing aircraft with rudders. It's just that it doesn't use conventional boat style rudders attached to vertical stabilisers since it doesn't have any vertical stabilisers. Instead it uses drag rudders.

There are several different types of drag rudders. The B2 uses one of the most effective designs as proven by countless experiments: the split elevon (sometimes called duckerons since they look sort of like duck bills). If you look at pictures of B2 landing you can clearly see them open at the wingtips.

Regardless of the type of drag rudders, they work via 2 principles:


The simple to understand one whereby if you increase the drag on one wing you make it advance slower than the other wing thus inducing a yaw.
The less obvious principle of shifting the center of drag backwards. This is the same principle of how badminton shuttlecocks work. Indeed, this is the same principle of how vertical stabilizers work. If the center of gravity is in front of the center of drag you get a weathervaning effect where the draggy part would swivel to the rear. The B2 achieves this by opening both drag rudders. In effect the B2 is creating a less efficient, more draggy, virtual vertical stabilizer. In practice you'd only need this extra stability when flying slow. At cruise speeds the drag rudders would be mostly closed (especially if you have them gyro stabilized, if you don't have computer controls and gyros you'd keep them partially open for stability).


Wing twist/washout

Some flying wings don't have drag rudders. Instead the wing is twisted to achieve a lift distribution that would cause the wing to have proverse yaw. The Dunne series of flying wings were promoted as inherently stable aircraft because they didn't experience adverse yaw. If you have proverse yaw you can simply use your ailerons to turn the plane without rudders.

The big disadvantage designs that don't use drag rudders or any vertical stabilizers is that you depend on banking in order to yaw. This makes landing on straight runways impractical so most such planes land on grass fields which allows the pilot to choose landing direction so that crosswind is not an issue. A lot of Dunne flying wings were seaplanes.