He might also get angry. Peter has a strong sense of justice. If he feels that the Avengers are being unfair, he could stand up to them. He'll use his quick wit and his powers to show that he's not going to be pushed around easily.
Tony Stark might use his quick wit and technology. He could immediately scan Peter to make sure there's no physical danger causing the hyperventilation. Then he'd probably crack a joke to try and calm Peter down, like 'Kid, you breathe any faster and you'll fly away like a human jetpack.'
Peter is likely to be in awe. He might be a bit nervous at first, but also extremely excited. He has looked up to the Avengers for a long time, so meeting them would be like a dream come true. He would probably be stuttering a bit as he tries to introduce himself and show his respect.
Peter might be very defensive at first. He could argue back, trying to explain his actions. After all, he is a teenager with his own ideas and he may think that the Avengers don't fully understand his perspective.
The Avengers would initially be in shock when they find out Peter is sick. Clint Barton would be the one to keep a calm head and start organizing things. Wanda Maximoff would try to use her powers to ease Peter's pain if possible. Vision would use his vast knowledge to try and find a solution. They would all be very concerned and do everything they can to make Peter better, from getting the best medical help to providing a safe and comfortable environment for him to recover.
The Avengers might react in different ways. Steve Rogers would probably be polite but firm in getting to know Peter's boyfriend. He'd want to ensure that he was a good influence on Peter. Thor, on the other hand, might be a bit more boisterous. He could either be really friendly and enthusiastic, or his loud and powerful presence might accidentally scare the boyfriend a bit.
They might be a bit surprised at first. Maybe Tony makes some jokes about British tea and bad teeth, but in a friendly way. Steve would probably be intrigued by Peter's different cultural background and might ask him about British history.
Natasha Romanoff could show her emotions in a more subtle way. She might look at Peter's old things and a small smile would cross her face as she remembers his innocence and determination. Thor, on the other hand, might become more determined to protect the world in Peter's honor, as he respected Peter's strength despite his young age.
They might be shocked at some of the dangerous situations Peter has been in without their knowledge. They could feel guilty for not being there to protect him more. And they would likely be very proud of his selfless acts.
Peter might be really nervous at first. He's used to being just the friendly neighborhood Spider - Man, and now the Avengers know he's Tony's son. He could be worried about living up to their expectations.