Every summer I try to shop for a staple sleeveless dress to make the hot days bearable. To my dismay and disappointment the armholes, more often than not, are almost to my waist and most of my bra is showing. When i first became cocky with my sewing skills I had no qualms about buying an off the rack garment with armhole issues, certain that I could fix it. Alas, these projects were often put to the side as an answer really did not exist.
Why you cannot fix the armhole level part 1: hence the name, a hole is a cut out. You cannot add back to something that is already final and therefore cannot fill in the underarm area. The only exception would be the addition of an inside gusset, but this generally screams bad alteration as wormanship is usually not done using same construction as original garment. Also, finding the same fabric in the same color is usually a difficult task.
Why you cannot fix the armhole level part 2: the remaining option is to pick up from the shoulder seams. This can be complex when you take facings, and armhole bindings into consideration as any excess uptake will now create extra bulk at the shoulder. Also, you must consider the neckline. By picking up at the shoulder seams, this also changes where the neckline will fall. Perhaps this helps out a neckline that is already too low, but it should be taken into consideration.
Should you or should you not? I would say to give it a try only if it is a thin delicate fabric and the extra bulk at the shoulder can be managed. I would only add the gusset if you can pass it off as a design detail and it looks like it belongs there.