A new and original antagonist could be a rogue psychologist who has developed a powerful hypnosis technique. This character might be jealous of Peter's abilities and want to use hypnosis to take over his mind and use his powers for their own gain. Or it could be a secret organization that wants to use Peter as a weapon, and they use hypnosis as a means of control.
It could also be an alien entity. Maybe an alien race has a form of hypnosis that they use on Peter when he stumbles upon their hidden base or something. They want to study his powers or use him to open a portal to their world. This would add an interesting sci - fi element to the fanfiction.
One possible antagonist could be a new super - villain who has a grudge against Peter Parker and targets his school to get to him.
The Green Goblin could be a main antagonist. He's known for causing chaos in New York City, and a building collapse would be right up his alley. He could be targeting the building for some strategic reason, like trying to get to Peter Parker or to create general mayhem.
Maybe it's a group of intergalactic criminals who are after the power that Peter Parker has as a 'child of the universe'. They could be stealing artifacts from different planets to gain more power and they see Peter as the ultimate source of power that they need to complete their collection.
The Green Goblin could be a main antagonist. He might see Peter's control of Alchemax as a threat to his own criminal empire and try to take over the company or destroy it. Another could be Doctor Octopus. With his brilliant mind, he may want to use Alchemax's resources for his own mad experiments.
The main antagonists could be the official Avengers themselves. Since Peter Parker has joined the rogue Avengers, the main Avengers group might see them as a threat to the order they are trying to maintain. They could try to stop the rogue Avengers from achieving their goals, which would put them in direct conflict with Peter and his new allies.
Often, other symbiotes can be the main antagonists. For instance, Venom himself might see Anti - Venom (Peter) as a threat and constantly try to attack him. Venom could be jealous of the power that Anti - Venom has or simply view Peter as an enemy because of the different nature of their symbiotic relationships.
Doctor Octopus could also work well. He might have a vendetta against the school because of some past incident. He could use his mechanical arms to attack the school, and Peter Parker would have to lead the defense. His intelligence and his ability to control those arms make him a formidable foe.
It could be a vengeful criminal that Spider - Man has put away in the past. This criminal has spent years in prison plotting his revenge and finally gets the chance when Peter is vulnerable. Maybe he manages to ambush Peter in an alleyway while he's on his way home from school or work.
One of the common antagonists could be Hydra agents. Since Steve has a long history with Hydra, it makes sense for them to be a threat in these fanfictions. They might be trying to steal some new technology that Peter has access to, or they could be after Steve again. Another antagonist could be some of Spider - Man's classic villains like the Green Goblin or Doctor Octopus. They might team up with other villains or forces to take on Steve and Peter together.
In some fanfictions, Zemo - the one who originally stirred up trouble in the movie Civil War - could still be a major antagonist. He might target Peter specifically to get to Tony or use Peter's inexperience against the heroes.