Because Nat asked him to because of what he was doing over the world. This is made explicit in the scene where Nat and Rhodey are talking and she does exactly that. Of course it's worth noting that Rhodey could also be acting off of his own back here and he might have been initially but he seems somewhat reluctant to which indicates that Nat was the one asking him to all along.
Natasha Romanoff: Where are you?
James Rhodes: Mexico. The federales found a room full of bodies. Looks like a bunch of cartel guys. Never even had the chance to get their guns off.
Natasha Romanoff: It's probably a rival gang.
James Rhodes: Except it isn't. It's definitely Barton. What he's done here, what he's been doing for the last few years... I mean, the scene that he left... I gotta tell you, there's a part of me that doesn't even want to find him.
Natasha Romanoff: Will you find out where he's going next?
James Rhodes: Nat...
Natasha Romanoff: Please.
James Rhodes: Okay.
There is further evidence that Nat is the one in charge and controlling clean up in that she is staying at HQ and seems to be coordinating efforts between the Avengers.
Natasha Romanoff: Alright. Uh, well. This channel is always active. So, if anything goes sideways... anyone's making trouble where they shouldn't... comes through me.
They are clearly following him because of his quest of murder and destruction to either try and bring him in and reason with him or to stop him altogether. When Nat finally goes to find him herself this is exactly what she does, she reasons with him and essentially gives him a bit of a slap on the wrist.
Clint Barton: You shouldn't be here.
Natasha Romanoff: Neither should you.
Clint Barton: I've got a job to do.
Natasha Romanoff: Is that what you're calling this? Killing all these people isn't gonna bring your family back.
No was he forgiven at the end? I wouldn't blankly state he was forgiven for what he did during the Blip but he did a lot to make up for it. He was a vital member during the Time Heist and he helped defeat Thanos' forces in the Battle of Earth, he more than made up for it./ Of course with losing Nat and everything else that happened it's hardly a leap to assume that quite a lot of it has been swept under the rug as there have been bigger things to worry about now.