Is it possible to use armatures with weight paint to shape a mesh permanently?

You can deform the mesh permanently by applying the Armature modifier on the mesh object.

If you want to retain the non-deformed version, you can duplicate with Shift+D, hide the original as a backup (H), then apply the modifier. It's handy to make a "trash" collection for cases like this; You can use the M key instead to send backup objects to the hidden collection in case you ever need them again.

It's also possible to edit the deformed mesh while posed without applying the modifier, by selecting "Display modifier in Edit mode" and "Adjust edit cage to modifier result" in the Armature modifier's options.

Lastly, you can apply the armature modifier as a Shape Key, couple options to try and see what works best for your task.

All modifiers can be applied with the apply button. Make a copy of your file for archive. Press the apply button on the armature modifier. You may need to apply more than one modifier.