I have a Stack Overflow app for iOS that doesn't use the API. It just uses search and web (scraping?) to display questions.
Do I need to get approval the from Stack Overflow team?
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You don't explicitly need permission, but you must follow certain rules and legal requirements.
We regularly block (IP range ban) unknown scrapers that do not identify themselves and/or have poor behavior patterns. These bans are permanent until someone emails us to make a case that they should be removed.
Follow the 5 guidelines that Jeff lays out in that answer.
All Applications must ensure they visually indicate that the Stack Exchange Network is the source of the content provided through the API Services (section 3a of the Stack Exchange Terms of Service). Applications which may in any way be indexed by search engines must adhere to all Attribution rules defined in the Stack Exchange Terms of Service. For applications that lack access to web browsing services, simple text versions of URLs (instead of hyperlinks) will be sufficient. Exceptions to these attribution rules must be requested in advance of deploying any Applications.
You must always abide by the the general terms of service and the Stack Exchange Network Content Policy. But as long as you don't plagiarize, harass, spam, fake accounts, try to cheat the system, or break laws, you should be good.