Yes, it’s possible. Select the mesh in object mode and in the modifier panel look to the armature modifier. To the right of the name of the modifier there are 3/4 buttons. One looks like a cube with 4 dots on the corners, click on that one, (hover over with mouse for a tool tip for what it does). After that, or if that button is on, there will be a 4th button, looks like an upside down triangle with 3 dots, one on each corner, enable that as well. The first button I mentioned enables the modifier while in edit mode, the second button applies the modifier to the mesh while in edit mode. Pose the armature and enter edit mode for the mesh and play with those two buttons to understand what they do, should be easy to understand.
Additionally, when working with shapekeys, first you create a basis shapekey, then you create another shapekey that is a duplicate of that basis key. This is the one you will want to work on. You need to set it’s influence to 1, then to edit the actual shapekey there is a button in the shapekey panel that looks like the cube with 4 dots mentioned above, you need to enable that so the shapekey will be active while in edit mode so you can edit it’s vertices.
Two other thoughts:
Right now I understand that you need to be in edit mode to make vertice changes record into a shape key.
You can make changes to the shapekey in edit mode, or as you also stated, you can also make changes to the shapekey while in sculpt mode as well. Just make sure the shapekey’s influence slider is set to 1.
Second, the deformation of the armpits/side of the chest shown in your picture looks like it could be corrected by just weight painting. The parts of the sides of the chest that are bulging out while the arms are posed as you have them need to be weighted more to the chest bone and less to the upper arm bones.
If you’ve just used automatic weighting on the mesh, you’ll need to manually adjust it via weight painting. Automatic weighting, at best, only gives you a starting point and will need to be refined. Personally, I never use auto weights as I don’t think it does a good job and rather than clean up the mess it creates, I’d rather just start from scratch.