Start with a clean zone-in. Have the necessary item in your inventory, but not on. If you don't have any of them, simply change to MNK and use a robe, they're cheapest.
Talk to Angelica. After she's done, stand there and count 10 seconds.
Talk to Angelica again. Now she tells you to wear your item. With no hesitation, put item on.
Talk to her the last time, and you've completed the quest after long cutscene.
When you refuse to model and/or you are coming too late, you get the title "Lower than the Lowest Tunnelworm" instead of "Supermodel"
If you're not wearing a body piece you may be unable to get this quest, it seems. I was able to get it after equipping one. Then you talk to her to initiate. Wait 10 seconds. Talk again. THEN change into a robe. Or whatever your jobs equipment might be.
This quest is timed. If you take too long to come back to this character dressed in the requested item, she will tell you that you are "so yesterday's style" and the quest will end without success.
If you take too long, change back into your regular gear, zone, then zone back in and then talk to her twice to redo the quest.