Nikon D90 stuck on manual focus mode.

Good morning. You have likely already gotten an answer to this, but yes, it will.

Unlike less expensive models such as the D40, D60, D3000 and D5000, the D90 has a built in autofocus motor, which means that all Nikon F-mount autofocus-lenses (except two for the rare Nikon F3AF) can be used in autofocus mode for all newer generation autofocus lenses like the Tamron 200-800. However, you may want to look around a bit, as Tamron has come out with a new 150-600 lens that is simply terrific (I know, I own one) and the Big Sig from Sigma is equally great.

FYI

Good luck.

Does it shoot in manual focus? In which autofocus mode(s) does it not shoot? Are you trying to focus too close, or on a subject with insufficient contrast?

The D90 has three different autofocus modes. In the AF-S mode it will not shoot until it has focused. In the AF-C mode it will shoot when you press the shutter release button even if has not focused. In the AF-A mode the camera switches between the AF-S and AF-C modes so either can apply.Also, bear in mind that the camera doesn't really know what you're taking a picture of. It may focus on something other than your subject. If you're in the single-area AF mode then you control which of the eleven focus points the camera uses. In any of the dynamic-area AF modes the camera decides.For full details on how the various modes work, please consult your manual. You might also check out the Nikon D90 video tutorial.

If the gears are stripped, it will require an experienced tech to repair.

That being said, there should be a small switch near the lens mount to switch between auto and manual focus mode's, be sure that that switch is pointing to A. When set to M, the little screw recesses into the mount so it doesn't try to engage, when set to A, it'll extend very slightly.

Also, Nikon has manufactured two types of AF lenses - with the motor in the body and with the motor in the lens, as with most newer lenses. Your camera should have a small screw on the lens mount that drives the focus motor between the lens and the camera, but I'm not 100% certain on that. Your camera can focus lenses with and without this type of coupling. BUT, if that screw/motor in the body is not functioning, you'll only be able to use lenses which contain the focus motor inside them, or manual focus lenses (you can manually focus AF lenses too).

You probably have a problem with the A/M switch on the body-- flipping that switch should extend or contract the autofocus pin that drives the lens. This particular lens doesn't have its own focus motor, so there's no way that it could be focusing on its own without the pin from the body driving it.

Try taking the lens off of the camera and looking at the autofocus pin as you flip the switch from auto to manual focus.

Your camera may be faulty. Or you might not be "in focus". The D90 in any of the auto modes won't take a picture if the 'in-focus indicator' isn't on. If that indicator is blinking, then it means that it is not able to focus on the subject. It's all in the user manual.

Your motor may have gone bad. Try sending it to PhotoTech Repair Service in NYC. They are Nikon authorized, and can do the repair under warranty if you still have it. Also if you join they're facebook page, they will give you a 10% discount.
Here is the facebook page link, it has everything you need to send it in.
http://www.facebook.com/pages/New-York-NY/PhotoTech-Repair-Service-Inc/102527215194

Try a reset (hold down both buttons with green dots next to them) and see if that helps.

Also try cleaning the contacts on the lens itself (light rub with a pencil eraser usually works well).

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.