Will the Nikon Normal AF Nikkor 50mm f/1.8D Autofocus work properly on the D7000?

I tried to order this from an online retailer. They told me it wouldn't work on the D7000. They said I would have to buy the G which carries a much higher sales price.

Yes.

It won't autofocus with the D3100 or D5100, but it will with the D7000.

I have that very same lens, and it autofocuses fine on my D90 - which is the predecessor to the D7000.

It would and it is great.

What a retailer. Have you thought about shopping somewhere else? They should know things like these.

It will work fine, your D7000 has a motor inside of it for focusing. Lower bodies don't, so they need lenses that have AF-S (the motor). You will be fine.