Learn Turkish: Gecikmeden dolayı yolcular kızıyor. - The passengers are getting angry because of the delay.

The postposition dolayı (because of / due to) requires the preceding word to be in the ablative case (-den / -dan). Therefore, the noun gecikme (delay) becomes gecikmeden dolayı.

Normally the verb kızmak takes the dative case. Why isn't there a dative word like gecikmeye in this sentence?

While kızmak takes the dative to show the target of the anger (who or what they are mad at), this sentence emphasizes the cause of the anger using -den dolayı (because of). They are getting angry because of the delay. If you said gecikmeye kızıyor, it would mean they are getting angry directly at the delay. Both are grammatically correct but have a slight difference in focus.

Is gecikme a standard noun or is it derived from a verb?

It is a verbal noun. It is formed by adding the -me suffix to the verb stem gecik- (to be late / to be delayed). While it acts as a gerund meaning 'the act of being delayed', gecikme has become fully lexicalized and is the standard, everyday noun for 'a delay'.

Since yolcular (passengers) is plural, shouldn't the verb be kızıyorlar?