There are 5 regional chests on Mercury. They can be picked up once per character. Opening all 5 gets you an emblem, 5 Mercury tokens, and 2-3 legendary items.

Locations for all 5 chests:

Chest 1

In the Lighthouse, there are 5 books with glowing crosses on them around the area. These have to be pressed in the correct order and can be acted upon by pressing your action button (you will be prompted)

1. On the table to the left of Brother Vance
2. Behind the 2 other disciples on the right side of the room, on a shelf
3. All the way to the left side of the left bookshelves
4. Almost at the far end of the 2 disciples' table, standing up between a stack of other books
5. On the upper half of the bookshelves to the right of the weapons forge. There's a ledge you can stand on to get it.

After this, you can head to the trenched area to the right of Brother Vance, where a conflux will appear. Use your action button on it, and it will open up a little doorway in the opposite trenches that contains the first chest.

Chest 2

At the area where you spawn in when you travel to Mercury, turn around, and take the path that leads to the back of the Lighthouse(not inside, but in the actual patrol area). There will be a Colossus guarding the chest. You can kill him and get the chest or get the chest and not kill him. But those stun rockets are too annoying and overpowered to move past.

Chest 3

To the right of the entrance to the Infinite forest, there is a bunch of Vex plating on the wall that is scale able. It's pretty hard, but when you get up far enough you can see a ledge up to the left you can try to jump up and grab it. It's a little tough. Fair warning.

Chest 4 and 5

Chest four and five can be activated in the same way, but require you to make the public event heroic, and unless you do something like stall on one side of the event, you may have to wait for the next event. They are located on both of the islands you can get boosted to.

Whatever island you get boosted to first is the only island you can activate heroic on. Do the normal steps you can do to activate heroic ( First video that came up on google) and when you reach the top, you can find the chest up there.

Keep in mind whoever puts the arc charges in last on their side is which island you get launched to, so choose your strategy carefully!! For those who didn't see the beginning, you can get an wmblem, legendary loot, and Mercury coins for getting the five chests.

(Taken from https://www.reddit.com/r/LowSodiumDestiny/comments/7ijroc/mercury_regional_chest_guide/).

I haven't tried doing it with only 4 but with my experience doing it normal I don't believe it's possible. I've tried doing it multiple times after one person has fallen off and keep going with only 5 and it will end up stopping and you can't continue.

There are multiple videos showing how it's possible to skip parts or to do it solo, by using swords and supers to jump the gaps between reactors, but these will only get you to the last area with nothing happening. You have to get to the last plate on each section and have the "reactor temperature rises" message appear for each to actually progress and have the enemies spawn at the final area.