Doctor's Orders (Story Mission)
This story mission becomes available after the freeing of Hope County Jail. Before you can start it, you'll have to free its quest giver. Start by rescuing him (and several other hostages) from the Mastodon Geothermal Park.
Lindsey himself is being watched over by a flamethrower heavy. As a plot character, he could probably survive the explosion, but why take chances.
Thoroughly scan the park, then snipe any hostiles that aren't being watched by anyone else.
This will clear a path for you to sneak up behind, and safely takedown the heavy.
Talk to Lindsey to begin the mission proper.
The first step is to dive down and get his bag. Despite appearances, the water isn't harmful. Dive in and retrieve the bag from a small cave.
When you surface, any remaining hostiles will attack. If you've already iced all the hostiles, only a helicopter will attack. Either way, deal with the threat before talking to Lindsey again. If he's interrupted during his spiel, it can bug the quest.
If everything goes smoothly, he'll task you to get two undamaged pronghorn skins and two undamaged black bear skins. If he gives his speech but your quest objective still says you have to return his bag to him, try fast traveling away and back a few times, then speak with him again.
The doc himself gives you a pronghorn hunting location. Search for the black bear skin on the ridge south and a little west of Hope County Jail.
For both, use arrows to avoid hide damage. Jess Black and other archer Guns for Hire can help out here. If you have the Harvest Master perk, you can use bullets as well. Lastly, while bait won't help with pronghorns, it'll certainly attract bears.
Meet up with Doc back in his office at the jail to complete this mission.