Professional teams have standby players. Hogwarts teams also have reserves, but their usage is inconsistent.
Harry's ex-captain, Oliver Wood, gets employed as a reserve.
Oliver Wood, the old captain of Harry's house Quidditch team, who had just left Hogwarts, dragged Harry over to his parents' tent to introduce him, and told him excitedly that he had just been signed to the Puddlemere United reserve team.
(Goblet of Fire, Chapter 7, Bagman and Crouch).
The presence of a reserve team points to a full squad of players with at least two senior teams. Presumably reserve players feature in the first team when there are long-term injuries or absences.
It should be noted that these standby players seem to be backup players which can start a game in place of another player if needed, rather than being in-game substitutes. As others have pointed out, Quidditch Through the Ages says that these substitutes are not permitted, which is backed up by Lynch and Krum carrying on even when badly injured during the Quidditch World Cup.
We do have one instance of a player being used as a reserve at Hogwarts.
"And she's really belting along up there, a neat pass to Alicia Spinnet, a good find of Oliver Wood's, last year only a reserve - back to Johnston...
(Philosopher's Stone, Chapter 11, Quidditch).
However, never see a consistent system of reserve players in place at Hogwarts. Alicia is the only named reserve that I'm aware of. We can only speculate about why Wood appointed a reserve player when that doesn't seem to be at all common at Hogwarts. It could be that Wood was particularly diligent in selecting standby players, more so than the other captains (although we never hear of him using them again). Or, more likely, Wood was simply spoilt for choice when it came to Chasers. Since he had so many good players at his disposal he chose to take the unusual step of naming a reserve player. He didn't have that luxury for the other positions. For instance, Harry states that there was no reserve Seeker.
Speaking quietly so that no one else would hear, Harry told the other two about Snape's sudden, sinister desire to be a Quidditch referee.
"Don't play," said Hermione at once.
"Say you're ill," said Ron.
"Pretend to break your leg," Hermione suggested.
"Really break your leg," said Ron.
"I can't," said Harry. "There isn't a reserve Seeker. If I back out, Gryffindor can't play at all."
(Philosopher's Stone, Chapter 13, Nicolas Flamel).
Why isn't the use of reserves more commonplace at Hogwarts? I would put this down to it being a school-level competition, without access to the resources or the talent pool of professional teams. Teams do replace players when need be but this seems to be done spontaneously and at the discretion of the captain. Dean Thomas was picked in this way.
He kept putting off replacing Katie in the hope that she would return, but their opening match against Slytherin was looming and he finally had to accept that she would not be back in time to play.
Harry did not think he could stand another full-house tryout. With a sinking feeling that had little to do with Quidditch, he cornered Dean Thomas after Transfiguration one day.
(Half-Blood Prince, Chapter 14, Felix Felicis).
As was Cormac McLaggen.
A closer look showed him that it was Cormac McLaggen.
"I've been waiting for you to come back," said McLaggen, disregarding Harry's drawn wand. "Must've fallen asleep. Look, I saw them taking Weasley up to the hospital wing earlier. Didn't look like he'll be fit for next week's match."
It took Harry a few moments to realise what McLaggen was talking about.
"Oh...right...Quidditch," he said, putting his wand back into the belt of his jeans and running a hand wearily through his hair. "Yeah...he might not make it."
"Well, then, I'll be playing Keeper, won't I?" said McLaggen.
"Yeah," said Harry. "Yeah, I suppose so..."
He could not think of an argument against it; after all, McLaggen had certainly performed second best in the trials.
(Half-Blood Prince, Chapter 19, Elf Tails).
As was Harper.
"Conditions look ideal," said Ginny, ignoring Ron. "And guess what? That Slytherin Chaser Vaisey - he took a Bludger in the head yesterday during their practice, and he's too sore to play! And even better than that - Malfoy's gone off sick too!"
"What?" said Harry, wheeling round to stare at her. "He's ill? What's wrong with him?"
"No idea, but it's great for us," said Ginny brightly. "They're playing Harper instead; he's in my year and he's an idiot."
(Half-Blood Prince, Chapter 14, Felix Felicis).
In the case of the game Harry missed Adamant credibly suggests that there wasn't sufficient time to find a replacement.
I think it's just a case of school-level sport being less organised and less regimented than professional sport, which isn't that surprising really.