Does a creature with vigilance still get to attack and keep its untapped status if it has summoning sickness?

Vigilance (CR 702.20) is a keyword ability that modifies how a creature attacks — specifically, it means the creature does not tap when declared as an attacker (CR 702.20a). However, vigilance does nothing to address whether a creature is eligible to attack in the first place.

A creature is subject to summoning sickness (the 'attack/activate' restriction) under CR 302.6, which states that a creature cannot attack, and cannot use activated abilities with the tap or untap symbol in their cost, unless it has been under its controller's control continuously since the beginning of that player's most recent turn. This restriction is separate from the tapping action that vigilance circumvents.

CR 508.1 lists the rules for declaring attackers. Under CR 508.1b, a creature cannot be declared as an attacker if it is affected by the CR 302.6 restriction (commonly called summoning sickness). Since the creature never legally becomes an attacker, vigilance never gets the chance to prevent it from tapping.

Concrete example: You play a Serra Angel (which has vigilance) on your turn. On that same turn, you cannot attack with it because it has summoning sickness. Come your next turn, you may attack with it freely, and vigilance will keep it untapped after attacking.

In short: vigilance and summoning sickness govern two entirely different steps. Summoning sickness gates whether you can attack; vigilance only modifies the tapping consequence of attacking. You need to pass the summoning sickness check before vigilance becomes relevant.