Can a creature with summoning sickness use a tap ability that does not have the tap symbol in its cost?

Summoning sickness is governed by CR 302.6, which states that a creature cannot attack or use an activated ability with the tap symbol (T) or untap symbol (Q) in its cost unless it has haste or has been under its controller's control since the start of their most recent turn.

However, this does not directly answer your question, because some abilities require you to tap the creature as part of their cost without using the actual tap symbol — for example, an ability that says 'Tap this creature: Do X' written in plain text. In practice, Wizards writes virtually all tap-requiring abilities using the official tap symbol, so a 'tap ability without the tap symbol' is extremely rare in modern cards.

If an ability's cost genuinely does not include the tap symbol and does not say to tap the creature, summoning sickness does NOT prevent its use. CR 302.6 only restricts the tap and untap symbols, nothing else. The creature can freely activate other abilities even the turn it enters the battlefield.

Concrete example: A creature with the ability '{2}: Draw a card' (no tap symbol) can use that ability the very turn it enters the battlefield. By contrast, '{T}: Draw a card' is completely locked out by summoning sickness until the creature has been under your control since your turn began.

So the answer depends on the ability: if the cost truly contains no tap or untap symbol and doesn't otherwise require tapping, summoning sickness is irrelevant and the ability may be activated freely (CR 302.6, 602.1).