How can I get the database of an app that only has a search field
"How can I get that database? "  The legal way would be to just ask them for it. Their database likely has passwords/etc which means it is not meant for public use outside of the functionality that the application you mentioned provides. Are there ways to access this data without asking the owner of the information? Possibly. But to attain the data in any other way then asking for it and receiving it, then my friend you are entering a grey area. A crawler won't be much help for what you're trying to do.
Here in Brazil, the government made an app that you can type in the name of the item, for example "Refined Sugar 2kg" and the app return to you the location of the nearest market that has the cheapest item.
It extends to all sort of products, and I'm thinking about creating an app using that database. It is public, so it's not a crime, but they do not list all the products on the screen right away with categories and etc. The app only have a search bar, and that's it.
How can I get that database? I thought of an crawler (probably python), but all the crawlers that i see has an webpage as a 'target'. I never programmed mobile, so I don't know if a crawler can do that job.
So, what should I study in order to make an tool to extract that information?