How to scrape data from inside Django app
Django follows MVT i.e Model (part where you write things related to the database ) , View (the logic analogous to what we did in controller - ref. Java) , Template(things that you'll actually see) . 
 As suggested by Alex you can have some inputs collected on your home page and using that data to scrape desired pages. 
Coming to your next question, yes you can launch the script on button click and basic working knowledge of JS would do good. 
This is like a very general answer synonymous to how general the question is so please feel free to get more specific requests if needed.
As an exercise, I came up with an idea of the following Django project: a web app with literally one button to scrape room data from Airbnb and one text area to display the retrieved data in a sorted manner.
Preferably, for scraping I would like to use Selenium, as there is no API for this page. So the button would somehow need to launch the browser automation.
So question number one is: is it possible to launch selenium from a web app?
Furthermore, I already have the working script for collecting the data, however I dont't know how to fit it in a Django project: models, views, separate script? My initial idea was to launch the scraping script on button click, then dump retrieved room-related data to database (updating model's Room attributes like "price" and "link" for example) and display the data back in the text area mentioned before. 
So question two is: is it possbile to launch Python script in a web app on button click, for example by nesting in a Django template? Or would other technologies be required, such as Javascript?
I know my question is general, but I am also looking for general advice, not a ready code sample. I am also open to other approach if what I just wrote doesn't make any sense.