file access without download

You are going to have to download the file in order to read it - that's the way the internet works.  Dropbox won't stream the data to you, you need a database server to do that, and that is an approach that your client seems to have outright rejected.

The only way you can approach this is to create a Dropbox app which has folder level access, so that its restricted to one folder in the clients Dropbox.  Have them put the file to be imported in that folder with a given file name.

Your app can then access this file via the Dropbox API - this will allow *you* to access the file but no one else (other than the other people who have full access to the account).

Your clients are basically trying to replicate the security of a database server without actually having a database server - this isn't going to work and I think you are in for a much harder time going forward as a result (speaking as a professional software developer).

Have you considered using SQL Server for your back end and allowing them access to the data in Excel by using Data Driven worksheets? 

https://support.office.com/en-gb/Article/Connect-a-SQL-Server-database-to-your-workbook-22c39d8d-5b60-4d7e-9d4b-ce6680d43bad