For me, the climax is the part in the novel where Andrea, the young assistant to Miranda Priestley at Runway magazine, decides to leave and live her own life. She does so in a burst of anger and publically, so she fears that her career as a writer is ruined, but finds that her outrage has made the papers and that many are willing to buy her freelance articles. Andrea also meets an editor with whom she becomes friendly (hinting that there may be the opportunity for future full-time work at her magazine) at a teen magazine. The editor tells Andrea that she also worked for Runway before editing her own magazine, so there is an instant bond of knowing what the two women have been through having slaved for Miranda.
Although there are other parts of the novel which are tense, they all help to build up to the final burst of high tension--Andrea's public outburst and leaving/being fired from her job.