That Vivitar will just be trigger only though and may well fire even at excessive shutter speeds (ie you won't get an exposure but you'll see it fire). With a Nikon CLS compatible flash or trigger on the shoe, on a D3300 it will lock out shutter speeds above x-sync as this camera cannot do HSS. However, unless I've missed something on the product page
http://www.vivitar.com/product/117/archive/328/sf-3000 is a very simple device and the camera has no idea what it is so may well fire it anyway. It's also almost certain it doesn't do TTL. Aiming it somewhere lighter on aperture priority will make the camera choose a faster shutter sped and if it was a shutter speed issue, it would be more likely to fail when aimed at brighter areas.
However! The flash doesn't do TTL and as Phil says - keep it simple when learning flash. Put the camera in manual, dial in an aperture, and a shutter speed of 1/200th or below. With the flash turned off, adjust the ISO and aperture to get an exposure of the ambient light that you like. Then turn the flash on, set it to manual on the camera and "normal" on the Vivi 3300,and well, accept what you get as it doesn't look like you can even adjust the power output on this flash. To adjust the flash exposure (say you want one stop less flash) stop down the aperture by one stop (3 clicks on most cameras), or reduce the ISO by one stop (halve it). This will affect the whole image, ambient and flash. Then, double the shutter duration to bring the ambient back up (this won't affect the flash exposure).
TBH, looking around the internet, lots of people have the same issue with this flash - Nikon, Canon and Olympus users all saying similar things. This coupled with the apparently random nature of the fault leads me to conclude it just doesn't work very well - ie whatever you do with the setup, it's going to fire 1 time out of 4. Maybe have a look at how the centre pin lines up, and give the contacts a clean, replace batteries etc, however I suspect it's a lost case - even if it did work, it'd be very frustrating to use a flash with no adjustment, or bounce - it's probably less useful than the pop-up on-camera, as that does full TTL. Try it out as a slave though - ie fires when the pop-up fires.
https://www.nikonforums.com/forums/topic/3256-external-flash-wont-consistently-fire/