I have a problem,

In my new campaign, I’ve asked the players to make backstories, so that I could tie them into the written plot ( I’m running Icespire peak ).
The first two players gave me their backstories almost as soon as I sent the message, so I took the first one, put it into the adventure, it was all good. The second one got a little more complicated. They had written a pretty detailed story, but somehow, it fit perfectly into the plot. I could almost replace a whole segment of the campaign and replace it with something from their backround. It was amazing. It fit so nicely with the setting too.

Now the issue :
The two other sent me backstories as well, and… they were ok. Nothing incredible, just your standard ‘ I am a social outcast, I wanna get revenge’. One of them, I gave a small part in the beginning, and since they’re a Paladin I get to mess around with their oath of they step out of line with their tenets ( don’t worry imma buff him to make it fair ).
Now I don’t want the other one feeling like I didn’t look at his backstory. I did. It’s just… it dosent fit nicely with the rest of the story. I know I could ask him to re write it, but that feels a bit scummy.

What should I do?

If the background/backstory doesn't fit the concept of the campaign, then feel free to explain the general concept to the player and see if they can improve it.

e.g. a dungeon delver or demon hunter background/story isn't going to necessarily get on well in DoIP.

This is a perfectly valid Session 0 issue for a DM to have, so don't feel bad. I also don't think there's anything wrong with asking a player to tweak their character in Session 0 or before they join a campaign. Nothing scummy about it. As a DM you have a prerogative to manage the flow of the story from beginning to end and this includes what kinds of character concepts you will allow into your game. An insidious spymaster who pulls on a complex web of political favors probably won't have a good time in a wilderness frontier campaign, after all.

The other way you can handle this is to manage player expectations. If you don't like the idea of asking the player to tweak their concept you can always just say,  "I would like to incorporate your back stories into the campaign, but I may not be able to do so with everyone's, just know that I'm trying my best and want all of you to have fun."

What Farling said. Nobody wants to play a character who doesn't really get to do the thing that they were made for. And not every character fits in every scenario. I think you got extremely lucky here, because it sounds like nobody was told anything and most of them managed to meet the brief regardless. Might as well elaborate a little so the player has a better idea of the vibe.

Also, I'm gonna take a side bar here and suggest that increasing the degree to which you as the DM hold power over the Paladin player's decisions is a very risky proposition. You know your group better than I do, I'm just saying if you're not certain your Paladin player is one of the minority who still prefers the "take away his toys if he's naughty" method, then you should check. By buffing him up you're basically increasing the amount of toys he can lose this way.

Also as I understand it this adventure requires some morally grey decisions.

Quote from ChoirOfFire >>

Also as I understand it this adventure requires some morally grey decisions.

 Not that I remember? There are a couple of choices, but nothing restrictive and generally the good choices pay off. Maybe I'm forgetting something.

Not everyone is good at writing backstories with a ton of material you, the DM, can utilize.  But that is okay!  Your job as DM is to take their sparse information and help flesh it out as the campaign progresses.  Even a single sentence backstory can be sufficient for your campaign's needs.  As the campaign progresses, ask your player outside of game details about their backstory--"hey, you escaped a cult, what did the cult worship?"--things that build off information you know and take it a step further.  You only need enough of their backstory at any given time to get through the next day's session.

If your player is not being forthcoming with their backstory, you can also help it along.  "You run into a cult of Tiamat.  You, Player A, recognize them as the cult that kidnapped you."  This is a last resort, since it means you are taking some agency away from the player, but it can be a good way to flesh out their backstory and get them thinking about it more.

Also, as a piece of tangential advice, the non sequitur backstory interlude is one of the worst things a DM can do to the campaign.  By that, I mean a side quest exclusively dedicated to a single player's backstory--if no one else at the table is invested in the quest, it can be really boring to go solve someone else's problem without doing anything to develop your own character.  Try to tie backstory side quests into the main plot.  "You have to go after the cult that kidnapped you because they have an item everyone in your party needs to complete the main objective."  You should also not resolve all the backstory issues at once, spreading it out in bits and chucks, interspersed throughout the campaign.  This keeps everyone's character invested in the plot and keeps you from spending too much time on one person.

One thing to consider too is maybe this character just isn't comfortable with that element.  They want to play D&D, they want to roleplay but they really don't know how to get there.  Just PLAYING the game for this person is something they need.

I think backstory really is one of those elements that is looked on way too heavily in games.  It's not to say its not important, and it's not to say it won't come up sometimes.  Most games are perfectly sufficient if they just can answer three basic questions.

Why are you here?

What's your main goal?

Why are you with the party?

If they can asnwer me those questions three, then great.  You'll do fine.  

Quote from Sergeant_Garlic >>

Now the issue :
The two other sent me backstories as well, and… they were ok. Nothing incredible, just your standard ‘ I am a social outcast, I wanna get revenge’. One of them, I gave a small part in the beginning, and since they’re a Paladin I get to mess around with their oath of they step out of line with their tenets ( don’t worry imma buff him to make it fair ).
Now I don’t want the other one feeling like I didn’t look at his backstory. I did. It’s just… it dosent fit nicely with the rest of the story. I know I could ask him to re write it, but that feels a bit scummy.

What should I do?

There is absolutely nothing wrong with a player coming into a campaign with a fairly sparse backstory and then filling in details as they go along, either via plot points that you introduce and they think are interesting, or even just via riffing and bouncing ideas off the other players

If what he has now isn't a great fit for what you have planned, just let him know and discuss whether he wants to add in details later to create some hooks as things occur to him, or simply not worry about it

1. Talk it out. The vast majority of issues with D&D can be resolved if either party involved brings up the subject politely and privately, but it's common advice that requires someone to act. And sometimes that's hard, they're not comfortable with it. This is an instance where your position as DM can help that player, by telling them further what you'd like for their game. Ultimately the game is for everyone at the table, you yourself being a player in some regard, and you want to have fun with it too. Let the player know you'd like to work with them, but don't rewrite their backstory otherwise they're not playing their character, they're playing your character.
2. Is it that important? I'm very much like the player in your group: I can write a backstory no problem, but I don't want one that puts the spotlight on me. I treat D&D as a 'beer and pretzels game,' meaning I'm there more for the social aspect than combat and roleplay. Don't get me wrong, my character doesn't shirk from their duties as a Cleric of Kelemvor, but that's the most in-character they want to be. Beyond that they're here for the ride, same as me. That isn't an incorrect way of playing the game, but it doesn't always mesh with other groups. I'm very lucky to have a group that understands my reasons for being shy and not that into having attention suddenly thrust upon them.
3. Shape up or ship out. If it's gamebreaking that this one player doesn't have a suitable backstory and isn't going to adjust for your game, it's OK to tell them that they may want to sit this one out. That may change your campaign up all of a sudden, with players deciding they don't want to play without that person, or would prefer a different module that the player in question can do. Or maybe they'll put some more effort in and want to cooperate with a backstory, but there's no point them staying if they do so begrudgingly, as that may come to disrupt the group when they're now given greater responsibility.

I hope these points, in addition to what everyone else has said, prove useful! ^^