Short answer: very, very rare.
Longer answer: not as rare as we’d like.
An older study of anaphylactic shock for other vaccines found five cases in almost eight million doses of vaccine given, which works out to less that one in a million chance of getting anaphylactic shock from other vaccines. (See full study here.)
A recently published study for influenza vaccination found the risk of anaphylactic was slightly higher. Slightly more than one in a million, but not quite two in a million. (Study here.)
The take home? Having an anaphylactic shock reaction after a vaccine is about one in a million. But that doesn’t mean it doesn’t happen, and it may be delayed. “The onset of symptoms among cases was within 30 minutes (8 cases), 30 to less than 120 minutes (8 cases), 2 to less than 4 hours (10 cases), 4 to 8 hours (2 cases), the next day (1 case), and not documented (4 cases).” (study above.)