Blue Vitriol Survey in the Mire
Start : Requisition Officer Reward: 1 Power, 200 Influence Materials : 20 Blue Vitriol, 10 Summer Stones
Depending on the amount of exploration you've been doing up to now, this may be very easy. 20 Summer Stones and 10 Blue Vitriol are needed here. Summer Stones are found near The Den landmark as well as pretty much everywhere (you can find them in Storm Coast as well). Blue Vitriol is throughout the Mire and really can't be missed.
Artifact Requisition in the Mire
This request hinges upon you finding one "Fereldan Medallion of Service" and one "Fereldan Tome". "Where the heck are these?" you may ask. Fair question! Corpse enemies have a rare chance of dropping both of these. This is likely added to their drop list once you have the quest, so head out and keep killing those corpses, but be sure to check what they leave behind. After 20-30 kills you will very likely have this quest completed.
You will get the "Summer Stone Survey in the Mire" quest after this one.
Elixir Requisition in the Mire
Start : Requisition Officer Reward : 1 Power, 200 Influence Materials : 5 Elfroot, 5 Diseased Tissue
Well, seems like you need elixirs to combat the plague! Elfroot is easy enough of course, but what about the diseased tissue? That is found out on corpses that are floating in the bog. Looting them will likely give you what you need.
Of course, the catch is that out in the bog you will likely be attacked by corpses, so be sure to clear out any enemies after looting. Once you have five pieces of flesh you can turn this in.
Note that you will likely get the "Artifact Requisition in the Mire" quest after turning this in.