Does anyone know whether the exception handling implementation supports restartable exceptions like Common Lisp's and Scheme's?
No, that functionality would fall under the stack-switching proposal, which builds on the tags of Wasm exception handling.
Speaking for CL, it seems so for me.
The whole magic about CL's condition system is to keep on executing code in the context of a given condition instead of immediately unwinding the stack, and this can be done if you control code generation.
Everything else necessary, including dynamic variables, can be implemented on top of a sane enough language with dynamic memory management - see https://github.com/phoe/cafe-latte for a whole condition system implemented in Java. You could probably reimplement a lot of this in WASM, which now has a unwind-to-this-location primitive.
Also see https://raw.githubusercontent.com/phoe-trash/meetings/master... for an earlier presentation of mine on the topic. "We need means of unwinding and «finally» blocks" is the key here.