Perhaps Harry was investigating something related to Voldemort on his own and got in over his head. He could have been captured by Voldemort's followers who were hiding him away. It's also possible that a magical spell backfired on him while he was practicing magic in a secluded area and it made him disappear, only to be found later by Snape.
Snape's reaction would likely be more complex than others. While others might show immediate relief, Snape might be more guarded. He has a history of animosity with Harry, so he might not show his true feelings right away. He could be more critical, even as he helps Harry, asking him how he got into such a mess in the first place.
One place to start looking for the missing fanfiction could be on large fanfiction platforms like FanFiction.net or Archive of Our Own. They have extensive collections of Harry Potter fanfictions. Check their search functions, maybe filter by characters (Snape and Harry Potter) and relevant tags.
Look for any unique plot details you remember. If it involves Snape finding Harry Potter in a particular place, like a hidden chamber or a different magical realm, use that as a search keyword. For example, if it was in a 'forbidden forest' setting, search for 'Snape finds Harry Potter in the Forbidden Forest fanfiction'.
One possibility could be that he has gone into hiding to protect a secret. Maybe he discovered some dark magic plot and has to keep himself away from the wrong people until he can figure out how to stop it.
If Snape goes missing, Hogwarts would be in chaos. His students would be worried and confused. The other teachers would have to take over his classes, but no one could replace his unique teaching style.
In this kind of fanfiction scenario, Snape might initially be conflicted. On one hand, he has his long - standing dislike for James Potter, Harry's father. But on the other, his sense of duty as a Hogwarts professor could kick in. He might reluctantly start to heal Harry, all the while muttering under his breath about how much trouble Harry always seems to get into, just like his father. Then, as he heals Harry, he might start to see some of Lily, Harry's mother, in him and soften a bit.
He would probably be in shock. Snape has spent so much time around Harry in one way or another throughout his years at Hogwarts. His immediate reaction could be disbelief. He might stand there frozen for a moment, unable to process what he's seeing. Then, the full weight of the situation would hit him, and he could break down in a rare display of raw emotion.
He could be secretly worried. Snape might use his knowledge of potions and dark magic to try and find a cure or at least a way to stop the bleeding. He may also be conflicted, as his past relationship with Harry and his loyalty to different sides in the wizarding world play a role in how much he wants to help Harry.