Contribution can be earned directly from completing quests, or by gaining amounts of EXP towards unlocking additional points through quest completion.
Starting out, do as many quests as you can and you'll be on your way to 50. By time I started at the camp, made it to Veila, and was on my way to Heidel I hit 50 points by doing the story missions and a solid amount of side missions found.
Don't forget to use the conversation minigame to try and unlock additional quests from NPCs which can net you nice bonuses if you have spare Work Energy to use.
**Not confirmed, but apparently you can earn Contribution Point XP through Cooking and Alchemy stations. This is done by receiving the by-products from the stuff you make, and then looking at the products some may offer Contribution XP if you turn them into the correct people.