The bite has to hit
From the basic rules on monsters, we can see the following description of the "hit" notation (emphasis mine):
Hit. Any damage dealt or other effects that occur as a result of an attack hitting a target are described after the "Hit" notation.
Since the cockatrice's Bite includes the saving throw to avoid being petrified after the hit notation, as part of the bite action (emphasis mine):
Bite. Melee Weapon Attack: +3 to hit, reach 5 ft., one creature. Hit: 3 (1d4 + 1) piercing damage, and the target must succeed on a DC 11 Constitution saving throw against being magically petrified. (...)
As this is all part of the same action, it requires the bite to hit for any effect to trigger that comes after the "Hit" notation, including the saving throw to avoid being petrified.
Roll20 seems to be erroneously separating the quote into two sentences which may have caused the confusion (though I should note that even with this text, the saving throw would still require a hit). The text on roll20 that you linked states:
Hit: (1d4 + 1) piercing damage. The target must succeed on a dc 11 constitution saving throw against being magically petrified.