ostlund

This plugin lets you insert a upload form on your pages so visitors can upload files to a Dropbox account.

Dropbox Upload Form

Inserts a upload form for visitors to upload files to a Dropbox account

acespritech

Are you looking for a way to create an upload form that automatically sends WordPress file uploads to your Dropbox? Using this Plugin you can easily integrate your Dropbox to your WordPress website. You can select maximum file size till your dropbox maximum upload size.
Features:- 
1 – Easy to connect form with Dropbox. 
2 – There is no file limit for upload. 
2 – Admin have short-code of form to display form in any page using it.
Shortcode:- [aspl_dropbox]

ASPL Dropbox File Upload

Another Best Plugin for Integrate Dropbox With Your Upload Form.

Dropbox Upload Form vs ASPL Dropbox File Upload