Q: how to get dropbox on iOS 5

DropBox needs IOS 7 to install.  If you have a iPad 1 (no camera) it can't be updated beyond IOS 5.1.1.  Have you tried using the DropBox.com website ?

You can get a compatible version of Dropbox for iOS 5.1.1

To do this, you'll need to download the current Dropbox App to iTunes on a computer.   With that done, go to the App Store App on your iPad and in the Purchased Tab, look for the Dropbox App. Tap on the cloud button to start the download.

It will likely tell you its not compatible, but offer to download a compatible version.

Install the latest compatible version of an app on an earlier version of iOS or OS X - Apple Support