We're in the middle of the World Series and my catcher got injured and will miss the rest of the series.
I have two players on the ML roster that are playoff eligible, but the game will not let me add them in the middle of a series.
So, I have to play short-handed and if my other player gets hurt I'm really in trouble?
What happens if three players get hurt, and I don't have enough to field 9 hitters?

Can't switch once a series starts. I always drop the number of pitchers I carry in the postseason, you're really noy going to need your 4th & 5th starters or bottom 2 relievers. I'll carry 11 pitchers and balance positional.

In all the years I've played I never faced this. I thought about dropping a pitcher and putting an extra PH type on the roster for the playoffs and decided not to in case of one of those 16 inning games wreaking havoc on my staff.
The problem is with catchers though...having an extra hitter is one thing and, if they're at least have reasonable fielding ratings can cover a number of positions. If you lose your catchers it's a significant problem.

I always carry 3 catchers in post season, or at least have an emergency one that's a positional players that might have 30ish pc. I think you're more apt to have an injury than a 15 inning game

I dunno I had a lot more long games than injuries this season, but I also tend to keep guys with higher health ratings (not that that seems to matter) and 20 Training. Anyway, from now on...I will carry a third catcher into the playoffs.

I think in-game fatigue is really important for pitchers, so I tend to run pretty high stamina pitching staffs. So high stamina that I always run 11 pitchers, and occasionally my 11th pitcher won't pitch in more than 5-6 games all season.

In the playoffs I'm about 50/50 between staying at 11 pitchers and going to 10. But my 15th position player will always be a pinch hit bat or a defensive replacement/pinch runner... I never carry a 3rd catcher unless that happens to be the extra bat. I rarely have post-season injuries, and the odds that I'd have both my catchers hurt seem very long.

Assuming you have 13-14 position players, the odds of having so many injuries in a single series that you can’t field a team of 9 have to be astronomically small.

To answer your question though, I assume the Sim would plug a random pitcher into a lineup spot if there weren’t enough to man every position. I think I’ve heard of that happening to fill an in-game injury in minor league games when people didn’t have enough guys on the roster. But that was probably 15 years ago that I read about that, so idk if the game would still react that way.

I’ve made the playoffs over 100 times, probably played at least 200 playoff series, and I don’t recall ever having more than 1-2 position player injuries in a single series. Some others mention carrying a third catcher, and I guess it can’t hurt. TBH I’ve never done that, and it’s never been a problem. I’m sure I’ve had a minor catcher injury here and there, but my second catcher has always been able to play the rest of the games.