You can't use the Bevel operator to bevel the edges of your Solidified plane, without applying the modifier, because the edges don't exist at the time the operator is invoked. You can use a Bevel modifier to do it, below Solidify on the stack.
The first bevel, rounding the corners in XY, could be done with an operator, and that may be better, allowing you to, say, inset the plane, to avoid the normals in the triangulation of the large n-gon interpolating badly over the edges into the thickness. However here we go with modifiers all the way, on a once-subdivided plane..
- A bevel of the corner vertices only, by vertex-bevel-weight:
- Solidify, with defaults..
- Bevel the rim, by angle, again, with default settings out of view..
- The resulting normals aren't perfect, but can be fixed with a Weighted Normal modifier.