Vivitar telephoto lense 500mm

At the risk of being obvious, you need a Nikon mount.

The lens comes with a T-mount. This is a "universal" mount with adapters for almost any camera mount, including the Nikon F mount. If the lens doesn't come with one, get a Nikon T-mount adapter.

The lens doesn't have any electronics, so you'll be shooting completely manual. No autofocus, no autoexposure, not even any exposure help from the camera's light meter.

For Nikon AF and manual focus SLRs. The compatible T Mount System makes it possible to use most any single lens reflex camera in combination with various front camera attachments such as telescopes, mirror lenses and slide duplicators. The T Mount is a metal connector that screws onto the compatible front attachment and allows this combination to be coupled to the SLR camera lens mount.

Assuming the lens has a Nikon mount, then yes. If it has a Minolta mount then it won't fit onto a Nikon camera.

Turn the aperture ring to its smallest opening (largest f/number). You can control the aperture from the camera body, the same way you do so on a lens without an aperture ring.

Vivitar makes lenses for a variety of cameras, so a lens for a Canon, for example, will not fit onto a Nikon. Assuming the Vivitar lens has a Nikon mount, then yes, it will mount on the D90 and yes, it will autofocus properly on the D90.

Check the Sigma site. They have the specs for the APO lenses.

it will mount fine, but I don't believe you will have autofocus.

Yes and yes. Bear in mind that not all features may work. You probably won't get autofocus or metering with the Nikon D90, for example.

You have named several excellent digital SLR cameras aimed at the top end of the amateur/semi professional market.
I have the Nikon D200 and have been very pleased with it.
As stated by JCDill, you must really decide what you want and can afford. The camera body you choose will commit you down one path or another. For example, if you choose Canon, only Canon or OEM (Other Equipment Manufacturers) such as Sigma lenses will fit your camera body. Canon have a good reputation for their auto-focus, while Nikon make lenses that are the envy of others.
If this is your first digital SLR then you might be best buying an entry level body with a multi-purpose zoom lens such as the Nikon D40. There is a lot to learn.
If you are converting from film SLR and already have a set of lenses, then the cameras you mentioned will probably be able to use your existing lenses (but this is less so with Canon than Nikon). I was ecstatic to disover that my old Nikon 500mm mirror lens works in manual mode (with metering displayed) on my Nikon D200 but would not work properly on my Nikon D70.
I hope this helps.

This lens is too old for the D90, it has not got the electronics for metering and autofocus.
You can use it manually in M mode but you will have to set the aperture and shutter yourself.
The D90 manual has a section on compatible lenses, page 228 in my version.
This recommends CPU lenses for the D90, type G and D.