You refer to
access logcat for an app
If "an app" is one that you are developing, then although Android now restricts installed apps from seeing the global logcat output, it still allows you to access data that your own process sends to logcat (whether from your own code, or from framework code that runs within your process). It's a simple matter of getting your app to run the logcat command on the device to send the data to a file, and then uploading that file to a webserver somewhere so you can read it.
(Read logcat programmatically within application gives the general principle, which is to exec the logcat program, and then read its output).
If however you are trying to get the logcat output from all the apps on the device, then getting adb working in some form is your only viable choice.
I assume you can install apps on the device, by placing them on an external website and downloading them using the device's webbrowser, then opening the downloaded file.
Thus you may be able to install a settings app extracted from a 4.2 phone, which might allow you to turn on adb debugging (though as you point out, the physical cabling is unusual).