Every time I've had them vaccinated I've generally ridden them that morning and the vet has come that afternoon...so not sure?
But they've all been ridden the day after
When I was young (many, many years ago) we were always told not to ride the pony that day and to take it steady for the next few days.
My vet told me that the new vaccines are quite different to the ones of old and that it is OK to ride the same day but not to overdo it or get the horse sweaty.