Go to your blog’s settings:
https://tumblr.com/settings/blog/**your-blog-name**
e.g. if my blog is glen-px, I would go to https://tumblr.com/settings/blog/**glen-px**
Scroll down to Submissions and Enable “Let people submit posts”:
You can set more options if you like, but the important thing is that the main toggle is enabled.
Wait a few minutes, then go to **your-blog-name**.tumblr.com/submit and you should see your submit box.