It could be one of his professors. If Teddy was constantly late to class or not doing his homework, a strict professor like Professor McGonagall might send a Howler to set him straight.
In the fanfiction, the Howler Teddy Lupin gets could be a case of mistaken identity. Some powerful magic could have misdirected the Howler meant for someone else to him. Teddy then has to figure out who the real recipient is while also dealing with the public humiliation of the Howler's scolding. He might start an investigation at Hogwarts, teaming up with his friends to solve the mystery and clear his name.
His mother could send the howler. In the wizarding world, mothers can be quite fierce when it comes to family reputation and she might be unhappy with Sirius's actions or friends.
It could be Severus Snape. He has a complex relationship with Dumbledore. Maybe Snape discovers some of Dumbledore's plans that he doesn't agree with regarding Harry Potter. He then sends a Howler in a fit of anger to express his disapproval.
Perhaps another student, like Hermione, could send the Howler in a very creative twist in the fanfiction. Maybe she has found out that Draco is planning something malicious against the Gryffindors and uses the Howler as a way to threaten him into stopping.
Hermione could be the sender. If Snape has made some unfair judgment in class against Gryffindor students, Hermione, with her strong sense of justice, might send a howler to him to voice her protest.
His mother is a likely sender. She could be worried about Remus and send a Howler to remind him to take his Wolfsbane Potion regularly or to be more cautious around others. Mothers are often protective, and in the wizarding world, a Howler is a rather forceful way to get your point across.
It could be Professor Snape. He often had a contentious relationship with Harry, and if Harry did something that Snape thought was particularly wrong or against school rules in a way that affected Snape's interests, he might send a Howler.
One possible scenario in this fanfiction could be that Teddy shows some unique magical abilities that remind Remus and Tonks of their own magical heritages. Maybe Teddy can transform in a way that combines both werewolf and metamorphmagus traits, which would be a really interesting plot point. It could also explore how Remus and Tonks, who had such a tragic end in the original story, get to experience some happy moments with their son Teddy through the magic of fanfiction.