The laws of tum'ah were not given until the Torah was given on Shavuos (or possibly later, depending on whether all mitzvos were given on Shavuos). Thus, even people who touched a corpse would not have been tamei and were able to bring the Korban Pesach at the correct time.
Note that in the verses about the laws of the first Korban Pesach, there is no mention that a tamei person can not bring it.