Whether or not one considers it to be naive on their part for believing they could remain anonymous on the internet is an entirely separate issue.
I think the site could use a new "modern" design, something that is more responsive.
I also think that the chat could be revised, it is often difficult with the numbers and leads to many "errors". A new, fresh design and implementation would not be bad here either.
The most straightforward way to do it would be to just have another field under each image for an instagram link, and then it would work more or less exactly the same as the current name field does. But this would mean that for any given person, you'd have to edit all of her pictures one by one to fill in the instagram link. This would be super tedious and it's unlikely people would go and backfill the links for all the existing pictures.
I take this opportunity to say that the GWM chat seems simply horrible and I really don't understand what utility it can have to justify the effort of its maintenance. I think it would be more practical to make it a separate page like this form. You could create a group in riot