A quality armless office chair will be difficult to find. You might try this route, go to http://www.chervan.com/. They are a KD frame maker that sells to the trade, small manufacturers and re-upholsterers. They make some office furniture frames that wouldnt be that difficult to assembly without the arms. Then you can have a custom finish and upholstered to match your desk. But you would have to go through a local shop or try their frame link. I hope this helps.

I saw some at IKEA recently, sorry I don't have any details.

I agree that it looks like a stress crack. The stress doesn't have to be from use of the furniture but can be within the metal itself. If this is not a major brand of patio furniture (you can get matching paint from the dealer) my personal approach would be to buy some acrylic/patio paint from a craft store, such as Michaels, and mix colors until you get something that matches your chair. Since the metal appears to be weathered, I would use a matte sealer on top of the acrylic paint (also sold at Michaels) to avoid a shiny repair spot. You may find you don't need any sealer, especially if the furniture is protected from the weather. The craft paints come in very small containers so picking up 2-3 colors to mix/match won't break the bank. Since the crack isn't on a joint I wouldn't worry too much about it. I think a tiny brush and enough paint to cover the exposed aluminum is all you need.

The trouble with making a fabric covering for a lawn chair is attaching it to the chair and getting a strong enough fabric to withstand the weight of a person sitting. They do sell webbing for chairs that can be covered with a cushion