I was able to complete this exact process using Tap Forms for iOS.
Once installed you can use an existing or newly created custom form to collect your data. Once complete you can export the file. The .csv will be available in the Export folder in the Files area. From there you can select the .csv and Upload to Dropbox. This will upload the .csv file to the root of your Dropbox.
Additionally you can open the .csv in Tap Forms and then use the share button (top right) to "Open In...", which will allow you to select the Dropbox app (assuming that you have it installed. Once Dropbox launches, you will be presented with the Dropbox upload process, which allows you to select the upload folder in your Dropbox account. You can also create a new folder as an upload destination.
Tap Forms also appears to support importing files from Dropbox as well.