Here is my 100% Achievement Guide for Deliver us The moon!
So this is a tricky one to explain.
A little bit into Chapter 5 (Reinhold Crater) You are tasked with taking out a dead canister (Or cell, whatever you prefer to call it) and replacing it with a new one (It is right next to you).
You then basically have to take this canister on a journey with you.
So keep going through the chapter as normal (Still scanning items, collecting collectibles, but ALWAYS take your canister with you).
You will soon come up to a console you need to press
on (For the doors to open).
From here, take a right and then down the stairs onto a sort of hangar bay.
Here you will see a hologram and an empty hole, so drop your canister down the hole (About middle of the park i would).
Grab the hologram, then directly behind you is a Moon Rover. Get in.
When you get put down onto the moon, reverse back and pick up the dead canister again, you now have 3 minutes to leg it down to the first tower (The closest one to you, and slightly to your right), Get down there, Drop it and leg it back to the Moon Rover.
Next, drive to the second tower furthest away from you (Towards your left), and complete that.
Now you have 3 minutes to complete the next section.
You start infront your moon rover, get it, and drive as fast as you can to the other tower (The one with the Dead canister by it).. Best advice is to go more towards the right, as you drive it round, that way you can drive it straight onto the pads without the need to reverse and straighten it out.
Once the door is open, grab the dead canister and VERY IMPORTANTLY - PUT THE DEAD CANISTER AT THE VERY REAR OF THE MOON ROVER (In the centre of it)
I'm not sure, but if you put it elsewhere it wont work.
Go and climb up the tower (It is collapsing, so you need to press
to jump over certain objects etc)
Once up the top and align the other tower, a cut scene starts and as long as you put the canister at the centre rear of the moon rover, this will unlock!
Of course, if you prefer to see a visual for this just go to 1:48:40 to see the start of it :)
Hope This Helps!