You need to disassemble the connecting parts to replace a broken spindle. It depends on where it is -- bow back, base, frame chair, etc.

If the spindle is straight and even, you can use a dowel rod. While you can buy these at any hardware store, you will have a better selection of species of wood at a woodworking store or hardwood dealer.

If the spindles are tapered or have beads and coves, you will need to have new ones turned. If you can find a local turner, you can have them locally made. Otherwise, the link below is a company that I've used. You send them a sample of what you want (for example, unbroken parts from another chair in the set) and they will put it on their duplicator machines and ship you back the original and replacements. I've found them very cost effective compared to a turner standing at a lathe.

Here is a link that might be useful: Normal Square Woodworking

Below is an irreverent look at chair repair.

Here is a link that might be useful: chair repair