I would like to extract crash logs of those users who are using my enterprise app. In case of the App Store apps I can get this information from iTunes connect. Since there is no intervention of App Store in case of Enterprise deployment therefore not sure how to access this information.
Is there any Objective-C API through which I can access the crash logs with the application code and upload them on my server using a web-service?
There is no Objective-C API available to access iOS generated crash reports. The only way to automatically access them, is adding a crash reporting framework to your apps that also catches the crashes and let you send them to a service or your own server.
I highly recommend not engineering your own crash reporting or exception handling mechanism, since it is way too complicated to make it right and not introduce more dangerous problems that could e.g. cause your app users to loose data.
The second part of my answer here lists a few options, contains open source projects, services with open source SDKs and closed source options: iOS crash log catch, debug info.. Catch and send via email to the Dev team