Would the nikon d3300 work well with the nikkor 70-300 af vr lens?

Would the nikon d3300 work well with the nikkor 70-300 af vr lens? - 1

There's a page in your manual to tell you, I suggest you read it.

We have to be very specific here. AF alone means it will not autofocus with the D3300 but luckily, there's no "Nikon 70-300mm AF VR" lens. There's a "70-300mm AF G" that will not autofocus with your camera and there's the "Nikon 1 70-300mm" that will not mount to your camera at all. Those with VR are the AF-S and AF-P versions. They will work well and autofocus with the D3300.

The Nikon AF-S 70-300 mm VR will work perfectly fine with your D3300 as will All Nikkor AF-S lenses

Yes, it will.