Elex Supplies is a side quest in Elex and it can be unlocked during your visit in Goliet village. You must go to the converter in Edan and search its insides for machine parts. Besides visiting the converter, you can also meet with Korin who received the same quest before you but didn't manage to complete it.
How to unlock this quest: Speak with Jora in Goliet.
Jora can be found inside of one of the huts Goliet. He can give you a quest during which you must go and find machine parts (if you have 2 skill points in Crafting you can show your knowledge about the process of extracting Elex during the conversation with him).
Your destination is the Converter in Edan located north-west from Goliet. There, you can find Albs and stronger monsters so don't start this mission if you don't have, at least, level 10. You can go straight to the converter or find a Berserker who was ordered to bring machine parts to Jora first. This person is called Korin and he hides near the converter. Talk with him about his journey for the parts. You can also promise him to retrieve his lost weapon – this is connected to another side quest called Necessary for Survival.
Start with finding entrance to the converter, it should be at the bottom of the construction. You can come across two Albs from one of the main quests, Contact – kill or avoid them. A Fighting Colossus mech is also near the converter but you don't have to fight with it. When you enter the converter, you must focus on reaching lifts that can take you to upper floors. Besides that, you must also deal with Threatening Rotboar monsters (mutated boars).
Machine Part can be found on one of the upper floors and an example of that place is presented in the picture above. Find three sets of parts and then exit the converter. This quest can be completed in two ways. The first variant is to bring them to Korin and allow him to take them to the village. The second option is to bring them to Jora in Goliet. Regardless of your choice, you get 850XP and Elexit (the amount of it depends on to who got the parts and whether you asked for payment before starting your searches).