How to setup a web app which can handle local data without uploading the data? Use python
If I get your point correctly, you want

Web connection, so your python program updated on server, client get it before using it.
Data store on local to avoid upload big file.

You can write a python program to check a server location to get your latest program if needed. You need a url / server file for program version / created date/time information to determine if you need to update or not.
After get latest python program, then start this python program to run locally.
With this said, What you need is to update your program to add below features:

Access your server, to get latest version information
Check against current version to see if you need to download latest program
Download latest version and use that to run locally.

Does this solve your problem?
I have a small python program to help my colleagues to analyse some tsv data. The data is not big, usually below 20MB. I developed the GUI with PyQT. I want to change this desktop program to a web app so my colleagues don't have to upgrade the program every time I change it or fix a bug. They can just go to a website and use Chrome as the GUI. 
So how do I do this? I spend most of my time developing desktop program and just know some basic web developing knowledges. I have read some framework such as flask and web2py, and know how to use HTML and Javascript to make buttons but no clue to achieve my purpose. 
Can someone give me a practical way to do this?
It'd be better if the user don't have to upload the local data to the server. Maybe just download the python code from server then execute in Chrome. Is this possible? 
Thanks.