There's a way to partially do this. It takes some tinkering.
There's a myth going around that Unity does not understand shape keys from Blender; this is not true. You'll find it under the "Blend Shapes" tag under Skinned Mesh Renderer. However, that's not everything, because the foundation of truth in that myth is that Unity does not understand Blender animations involving shape keys. You need to re-animate it from within Unity.
The additional concern is that shape keys always lerp, never slerp—that meaning, transitioning between shape A and B is done in an exclusively linear fashion. They also act in strictly Cartesian space, never spherical space (as nice as that would be). That means that for something like above, where you're "rolling up" a scroll, you'll need to break it into several parts.
I would suggest approaching something like this by breaking down the action into multiple individual segments, preferably atomic ones; for each one, develop a shape key that rolls the shape from one form to another. In Unity, you would maximize those one by one. You might have one shape key that forms the initial roll in the scroll; then another that half-rolls to the right, then another, until you have the entire scroll rolled up.
That is, have one shape key that goes from flat to flat-with-one-roll, then another that goes from flat-with-one-roll to flat-with-one-and-a-half-rolls, and continue until you reach the end. Then, in Unity, set up an animation that interpolates each Blend Shape from zero to one, then moves to the next one.
Conveniently, many of Blender's modifiers can apply themselves as shape keys.
I should also note that you'll want to add a solidify modifier to that mesh, even if at width zero. It looks like it's just a plane, and in Unity, one side will be transparent. I strongly suggest applying this after you've got the roll-up motion working, though; it should make things much easier.