I am very new to photography. I recently bought a Nikon D5200 NVR camera. Is it possible to replace its camera lens with a VR one or do I need to buy a new DSLR with VR lens on it?
Your camera is an interchangeable lens camera. You can buy other lenses with the Nikon mount and use any of them. That includes lenses with VR — vibration reduction — even if the lens which came with the camera does not have that feature.
Note that entry-level Nikon cameras like yours do not have focus motors built into the body, so assuming you want auto-focus to work, you do need to be aware of that. See What type of lens will auto-focus on Nikon bodies without a focus motor? for more. This does not affect VR, however. And, as far as I am aware, all lenses for Nikon which feature VR also have a focus motor — so while you should know this in general, it won't affect your current concern.
Also note that some companies — Pentax, Olympus, sometimes Sony — make interchangeable lens cameras where the image stabilization (another term for vibration reduction) is built into the camera body. Since you have a Nikon camera, this doesn't affect you, but it's worthwhile to be aware of this in general — if you bought a camera in one of those systems without VR, you would need a new camera. See What is the difference between in-lens image stabilizing and sensor-based image stabilizing? for more if you're interested.
The only things you really have to care about to buy lens for your camera are:
Obviously: Is it the right brand (Nikon, or another brand with a Nikon mount)?
Is it full-frame or APS-C? Your camera is an APS-C format, you can use full-frame lens on it, but they will essentially be more expensive for no benefit over APS-C lens.
Does it have built-in focus motor? Your camera does not have screw drive autofocus so you need built-in focus motor (AF-S in Nikon's jargon). Almost any recent lens has this.
As mattdm already said, VR does not come into the picture. It's internal to the lens and works with any camera.