Perhaps he has been exposed to some sort of deadly and unknown substance in the mythical world. This could be a new type of monster venom or a strange plant that he accidentally ingested. The substance could be attacking his body from the inside, causing his internal organs to bleed, and thus he throws up blood.
Grover would be panicked. He's Percy's friend and he cares a great deal about him. He might start running around trying to find help, like looking for a healer or someone with magical healing abilities. He'd probably be muttering to himself about how he should have protected Percy better.
Well, it could be due to some sort of magical potion gone wrong. Maybe he drank something that was supposed to give him extra strength but instead made his stomach turn. Another reason could be that he got poisoned during a battle with a monster. Some monsters might have venom or toxins that could make him sick to his stomach.
It could be related to a new and strange power he's trying to control. Percy is constantly evolving and getting new powers. If he tries to tap into a power that his body isn't ready for, it might have some adverse physical effects like vomiting. Like when he first started to understand his water - based powers, there could be something similar but more extreme with a new power.
It can create a sense of urgency. If Percy throws up, it might mean he's been poisoned or is seriously ill, and his friends will have to find a way to help him quickly. This can drive the plot forward as they search for a cure or a way to make him better.
Well, one reason could be that he was poisoned. In the world of Harry Potter, there are many poisonous substances. A fanfic writer might create a scenario where Harry is unknowingly poisoned, and this causes him to throw up blood as his body tries to fight off the toxin. Another possibility is that he has some sort of internal injury from a fight. If he got hit by a powerful spell or a physical attack during a battle, it could lead to internal bleeding which results in throwing up blood.
Regarding Annabeth, since she's also a demigod, her blood type could potentially be affected by her Athena - related heritage. Some fanfics might use this as a plot point, like her blood having a certain'smart' quality due to Athena's wisdom. But again, it's not a standard thing in the official Percy Jackson lore, so it varies a great deal from one fan - created story to another.
Well, in some fanfictions, it might be related to his demigod powers. Since he has a connection to the water element, blood being a liquid could potentially be manipulated in a way similar to how he controls water. Maybe he can sense the water content in blood and use his will to move or influence it.