I think we're going to need more information to help you... are you wanting to host your WordPress media on Dropbox, and then link to it?
I don't know how you could integrate this with the media/gallery function in WordPress but I would try something like this:
1. Upload your images to a subdirectory in your Dropbox "Public" folder.
2. Right click on the image and select "Copy Public Link"
3. Paste the copied link in the image URL field when inserting an image into a WordPress page/article.
But I'm curious why you would want to go to so much effort?
www.EasiestWebHosting.com - Keep it Simple.
● Web Hosting Should Be Easy! No technical jargon, no problems, no frustration.
● Low-cost shared hosting packages, optimized for small projects and WordPress blogs.