Vivitar Lens - d5300

That depends on what you mean by "make it work."

Vivitar made lenses in a wide variety of mounts, including Nikon F. This appears to be a Nikon F mount lens.

The lens will mount, but the camera cannot control any aspect of the imaging process because the D5300 does not have meter coupling/aperture indexing nor the ability to input non-cpu lens data.

It will be full manual exposure mode, manual aperture control by the lens ring, manual focus, and sunny 16 rule (or a handheld/secondary meter). There will also be no ability to use TTL flash for macro work. And because of the smaller apertures/reduced light and the new style focusing screen, I suspect manual focus will be quite problematic. Manual focus is always a problem for me (eyesight) and I do not think you will find the in-focus indicator accurate enough for critical/macro work... there is no adaptor that will change any of this.

It may be marginally easier to use with live-view.

That lens appears to have an F-mount, so it doesn’t require an adapter — it’s supposed to mount on Nikon SLRs.

But generally speaking, there are very few cases where you can adapt SLR lenses intended for other mounts to Nikon bodies. The flange focus distance (also called the registration distance) is deeper on Nikon SLRs than most other common 35mm format brands. Mounting a lens too far from the camera’s film or sensor plane prevents the lens from focusing at infinity, or some times even at usefully far distances.

For more on this, please see: Can I use lens brand X on interchangeable lens camera brand Y?