Yes, cat vaccinations can cause diarrhoea. Sometimes cats have an allergic reaction to a vaccine. These are called type I hypersensitivity responses or anaphylaxis. It is a well documented adverse response associated with vaccine administration in companion animals.
The hypersensitivity reaction usually occurs within minutes and shows in symptoms of diarrhoea, vomiting, upper respiratory issues such as sneezing and coughing. Other symptoms include anaphylactic shock, issues with the cardiovascular system, gastrointestinal tract, respiratory system and skin. There may be cardiovascular collapse. It appears that in very rare cases the reaction may prove fatal.
The treatments include the administration of anti-inflammatory agents such as antihistamines, corticosteroids or epinephrine. Cats suffering from an allergic reaction need to be monitored for at least 48 hours because a relapse might occur.
Source: Brooks R. Adverse reactions to canine and feline vaccines. Australian Veterinary Journal, 68:342-344 (1991) as referred to in Linda P Case’s The Cat.
Note: I am not a vet. I have a studied biology and am a careful researcher.
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Some more on cat vaccinations