Yes. It is also worth noting that dangerously severe allergic reactions to injectable substances start very, very fast. You'll begin to see it before you even leave the office. And your doctor will have the tools to reverse the reaction, in the vast majority of cases.
While especially extreme reactions do lead to death on extremely rare occasion (just like everything else in the world, from peanuts to aspirin), nearly all reactions can be treated on the spot.
Less severe reactions may take a couple hours, but you also would have plenty of time to get back to the doctor if need be, as the symptoms are slower and not as dangerous.
I have actually dealt with this myself. I had a cat who was severely allergic to something in vaccines. We were able to treat her immediately, and she was fine. This is the only reaction I have ever seen in any animal of any species -- it's rare. But yes, it happens. And she never got another vaccine again.
But because she was a cat and so many cats never get vaccines, she didn't benefit from herd immunity the way humans often do. And that meant I had to keep her quarantined from any other unknown cats for the rest of her life. Even if I pet a stray, I washed my hands and changed pants when I came in, because FeLuk can be caught even from saliva, and is invariably fatal once contracted.
If she'd been a human in a population of other humans, nearly all of whom were vaccinated, this wouldn't have been such a concern.
