Running Website Natively On Phone Apps

I have customers wanting my website in app form, but the website is responsive. I've been looking for an open source project or code to just run the website in an app or open up their default browser to direct them to it. They just want the button.

1. Is there an open source project to make it easy to develop that code for iOS, Android, etc.
2. Are there any issues with running a website natively? Aka, same as cross-browser issues or..?

Thanks!