Will a Vivitar lens work with my Nikon camera?

That is an FD lens which was made for Canon. You'll need to purchase an FD to F mount, which you should be able to find on ebay for less than $50.

I'd advise you get one with glass so you can continue to focus to infinity.

That is a Canon FD mount lens. While you could find an adapter to mount it on a D3200, it's problematic, even for adapting onto a Canon dSLR.

The main problem here is the registration distance. This is the distance that the lens is held by the mount away from the image plane (the sensor, in the case of a dSLR). This distance varies between mount systems, and lenses are designed specifically to work at this distance. If the mount you're adapting from has a thicker registration distance than the camera you own, adapting is easy--a simple ring can make up the distance as well as make the physical linkage possible.

However, when the mount you're adapting to is thicker than the one you're adapting from, you're in trouble. Because you can't just shove a lens farther into a camera body without damaging or modifying the physical linkage somehow. And the FD mount's registration distance is 42mm. Nikon F's is 46.5. That 4.5mm is in the wrong direction.

So, if you do find a simple adapter ring and use it, your lens will be sticking about about 6mm or more (because the ring adds to the depth) too far. Which means the lens won't be able to focus past a certain point. Probably measured in feet. It's like using a macro extension tube. While you gain in close focus ability, you lose in far focus ability. So you'll need a lens with a glass element in it to act like a teleconverter to regain focus-to-infinity. But that will increase the focal length, reduce the maximum aperture, and add softness. The cheaper that adapter is, the softer it's likely to make your lens.

In addition, the FD/FL mount was mechanical only. There's no electronic communication between the lens and the body as you'd have with a current Nikon lens. You can't autofocus. You have to set the aperture with the lens's aperture ring. Your D3200 cannot meter accurately with this lens. You can't shoot in any modes other than M and A, because the body can't tell the lens what aperture to use. And you won't have any EXIF information from the lens. In short, it'll be a pain in the ass to use.

And a Nikor 55-200 or 70-300 lens goes for around US$150 new on Amazon. And if $150 is too expensive for a lens for you, then a dSLR may not be the right type of camera for you. dSLRs are expensive because they're system cameras, and you have to buy the other parts of the system to take advantage of their versatility.