The earliest that you can go for this achievement is on Day 5, although you can do it on Day 6 if need be. You can possess 5 of the 6 enemies up until this point but you won't encounter your first Genie until your pretty far into Day 5 anyways. Keep in mind that others have reported that this must all be done on the same save file. I'm not certain if the progress transfers over from each day either, so I would aim for getting all 6 enemies possessed in the same day.
There are 6 different enemies that you need to possess with Demelza's Gauntlet.
1.) Golf Ball Furry: Golf balls that change into Brown monsters. Can be found around the wilderness and a few other areas, including the Golf Course. Make sure you possess one of these BEFORE progressing too far into the Quarry. I don't recall if there were any available in the Quarry/after you beat it.
2.) Silver Glove: Standard glove enemy. These can be found in almost any battle during the Quarry.
3.) Flaming Sword: Exactly as the name suggests. These can be found often with the Silver Glove enemies. They are very common in several parts throughout the Quarry.
4.) Possessed Refrigerator Magnet: These are the purple balls with the letters whirling around them. It can be tricky to get close enough to charge the Gauntlet and get your shot off before being damaged, but just keep trying. There are several of these in the Quarry.
5.) Blue/Flaming: These are the creepy Doll face balls that launch the fireball mortar's at you. Also can be tricky to get close enough, but not too bad. There are quite a few spots in the Quarry where you can find these.
6.) Genie: Genie with purple clouds. This one is interesting because one of the genies attacks sucks you into a vortex and deals damage with the clouds surrounding the Genie. You can always use Nessa's boombox to temporarily clear the clouds, then possess, if you're having troubles. I believe there are only 3 (maybe 4) of these enemies in the entire game. You'll come across your first Genie naturally as it is part of the story progression. There are 2 more after this so make sure you grab one early on to make it easy for yourself!
Icefiretn has confirmed that you do NOT need to possess a silver glove with the shield. So just possess the non-shielded ones and move on.
Keep in mind that the game has a pretty good autosave feature. That being said, if you someone don't possess an enemy that you need on your list, and it gets killed, just quickly load the last autosave and try again. Otherwise that persistent autosave might put you in a spot where you can go backwards, and then you risk having to start the day over again. This shouldn't be a huge concern, as there are multiple enemies of each type in this day, but be aware of it regardless.
If you think of anyways that I can improve this solution, please let me know.