A creature normally deals sneak attack damage to a foe that's vulnerable to sneak attack damage whenever the requirements for sneak attack are met and the creature makes a successful attack. It doesn't matter how many attacks the creature has; if the creature meets the requirements for dealing sneak attack damage, the sneak attack damage usually just happens.
However, broadly, to make more than one attack typically requires a creature to take a full-round action to make a full attack. This means the typical creature can't also move more than a 5 ft. (by taking a 5-ft. step) during that round. Thus foes that start the creature's turn adjacent to the creature will be in danger of experiencing multiple attacks from that creature's full attack, but foes 10 ft. or more away from the creature will continue their existence in relative safety. Whether these attacks are multiple natural attacks, multiple manufactured weapon attacks, or some combination of both also typically doesn't matter—a full attack's usually necessary to make more than one attack.
Further, some natural attacks—like a natural attack made with a tailblade during a full attack—are called by the game secondary natural attacks. Secondary natural attacks are made at a −5 penalty on the attack roll and deal their normal damage plus only half the creature's Strength bonus (instead of all of the creature's Strength bonus as with a primary attack).
Finally, damage reduction applies separately against each attack the creature makes. This means, for example, that a creature relying on a large number of little attacks that deal sneak attack damage may have difficulty dealing with a foe that possesses as little as DR 5/magic.
While four attacks that deal sneak attack damage can seem like a lot at low levels, wading into melee is usually really dangerous for the creature that deals sneak attack damage, and a foe's positioning can make it far more difficult to realize effectively all those attacks.