Who was the bomb seller?

The Apostles had the plutonium the entire time after the alley scene

Initially, John Lark and the Apostles were not working together. The Apostles were clients of Lark.

The White Widow had brokered a deal between the Apostles and John Lark for plutonium. Though due to the nature of this business, all communication is through the White Widow. Presumably through very secretive channels as not even she knows the identities of either party until the initial meeting between Lark and the Widow in Paris.

The apostles manage to steal the plutonium from Hunt in the alley at the beginning of the movie.

As payment for the plutonium, the Apostles demand John Lark break out Solomon Lane from custody. (Or in this case, Ethan pretending to be Lark).

After Solomon is freed, the Apostles give the plutonium to (the real) Lark for his plan in Kashmir in order to fulfill their part of the deal. It also seems at this point, Solomon agrees with this plan, so he decides to help Lark as a free agent.

I won't be completing backing up below theory, but that is the closest that I have currently found -

Based on Mission Impossible Fallout's Wiki page the plot section mentions the story. Let me break it to you,

• Intitally, in the alley scene, when Ethan saves Luther's life he looses the Plutonium Cores, which is later obtained by Apostles. (they get it from the alley)

• Later, at the fund raiser event, Ethan impersonates as John Lark, assuming that White Widow, the broker of the Plutonium Cores, haven't met John Lark ever.

• Here, reportedly, they are attacked by members of Apostles who tried to, kill both John Lark and White Widow, free Lane and acquire the Plutonium cores.

So to answer the question, members of Apostles or people hired from them, were the ones who were trying to kill Ethan and White Widow. A classic plot of one buyer trying to kill the broker, seller and get the item for free.

TBH, the plot gets quite messed up there, with people impersonating another, deals between buyer & seller, etc. etc.

I agree with DeeV it was the apostles and John Lark who setup the meeting with the White Widow and a fake John Lark knowing Ethan would take the fake John Larks place. It was all a ruse to make it look like Ethan was John Lark and frame Ethan. The apostles had no intention of giving away the plutonium. When Ethan got the mission at the beginning it explains how the apostles had already caused a smallpox outbreak in Kashmir. So they already started the scheme of luring Ethan’s x-wife there so she would be killed in the nuke.

Watched Three Times and here is What I infer.

Solomon Lane is the leader of Apostles. Lane never wanted nuclear explosions but Anarchy by other means.

It was Lark who wanted as drastic as a nuclear bomb.

This is clear by the fact that in the very beginning Lane (in Hunt's nightmare) warns Ethan that nuclear bomb is a FALLOUT of Lane's absence from Apostles. The absence of Lane has forced Lark to do Something drastic.

In the beginning in Berlin Apostles steal all the Three cores. Apostles strike deal with Lark for Release of Lane and frame Hunt. Lark will get the bombs to impose his will on the World. Apostles lend one plutonium to White Widow as an assurance to Lark to achieve his cooperation in freeing Lane.

But things to south, their plans in framing Hunt, Lark's collusion with Lane all get exposed after the Heist pulled by Hunt and his team. So Lark and Lane escape together to carry on explosions to be ultimately stopped by Ethan.