Is it possible to have a URL that will open an app on an iOS device, without the app being specially coded to handle that? Is there like a default url scheme, like openapp://Facebook, that would open the Facebook app (or whatever) if you click that URL from Safari on an iOS device? Or is there some other way to accomplish this?

And does the app have to be specially coded to handle this?

My company has an app but we don't have the budget right now to update it to handle any special URL schemes, but I'd really like a way to open it directly from a button on the web site.
The app have to be coded for it, and the server needs some configuration too.

Check this for more information: Support Universal Links.