In regards to directly answering your specific question, it is not possible to put an Apple device in Supervised mode without using Apple Configurator.app or having your device pre-configured with the DEP (for example from a company).
You can confirm this in this rather well-worded post, directly from Apple.
You've got some obvious work-arounds, using OSX in a VM or a friend's Mac, etc. In a comment above, someone recommended this app, which I cannot verify or authenticate.
Mobile phones are by nature, not secure at all. Even with very advanced, tailored solutions, for example, VMWare's air-watch, one cannot even verify they are secure. I would highly recommend readers to study the research by Karsten Noel, and others. In a positive sense, many of the features that make so-called smart phones, convenient, require them to continuously phone-home, whether this is through a 3rd party network like a VPN, or not.
If your main concern is to get a complete VPN solution, that is always-on, and routes 100% of your cellular and Wi-fi traffic, without a doubt, through a VPN-tunnel, I would suggest that is not possible at all through an iPhone, with normal hardware; because it's hard to tell without using Wireshark/
tcpdump or other tools on all the networks, to see what is happening when the phone comes off and on, or wakes up from sleep, etc., although there are several iOS apps, for example OpenVPN Connect.