Learn Turkish: Tutup onu azarlayamam; bende o cüret yok. - I can't just go and scold him; I don't have that audacity.

In this context, tutup (from tutmak) is acting as a converb of unexpected, abrupt, or audacious action. Rather than physical grabbing, it expresses suddenly taking it upon yourself to do something that might be out of line. It functions exactly like the English expressions "to just go and [do something]" or "to up and [do something]".

Could I use kalkıp instead of tutup in this sentence?

Yes, absolutely. Kalkıp (literally "getting up") is the other major converb used for audacious or unexpected actions. Kalkıp onu azarlayamam would mean the exact same thing: "I can't just up and scold him."

I remember that azarlamam means "I don't scold". How does the extra -ya- in azarlayamam change the meaning?

That -ya- (a buffer -y- plus the -a- ability suffix) adds the concept of capability. While azarlamam is the standard negative aorist ("I don't scold" or "I won't scold"), azarlayamam is the negative ability form ("I cannot scold" or "I don't have it in me to scold"). B2 learners often miss this small but crucial vowel difference in fast speech.

Why does the second half use bende ... yok (at me ... there is not) instead of saying ben ... sahip değilim (I do not possess)?