Which version of Moodle are you using? It is possible when setting up the essay question to allow attachments which then enables students to upload a file as an attachment.

If you are asking about students uploading pictures, no.

In Moodle 2.6.6 is it not possible for students to 'upload' images into Essay questions.

Students can only "insert" a picture into the textbox if they have the url, but they have no option to "Browse" in order to upload images into the quiz via the Insert Images button. The work around is to allow a file upload with the essay question.

I'd like students to be able to upload images in the Insert Image button option as well so they can upload a screen shot of their work for open ended math questions. It's much easier to grade open ended responses. 

As of now, I simply allow the open ended questions to allow a file attachment. The only problem with this method is that if students do not put any text into the textbox, the answer comes us as skipped, so it is marked with a zero by default (until the teacher makes the adjustment). To get around, this, I fill in text in the Response  Template so Moodle inserts text automatically and this question doesn't get marked as 'skipped'.