All you need is the equipment, and most of all, a lot of patience since you cannot predict where and when a bright meteor will occur. Just about any good camera from a 35 mm to a large format camera like a 4X5 press type can be used. You simply mount a prism or replica transmission diffraction grating in front of the lens, no slit is required. The dispersive element, prism or grating, breaks the light of the meteor up into a spectrum. Today, gratings are preferred over prisms as the dispersion is better and nearly linear.
Good gratings are expensive, however.
A large format camera is preferred if you want to obtain good spectra. You can sometimes find these lenses in used camera stores or on eBay. Once you have the equipment, you then start taking time exposures of the night sky hoping a very bright meteor crosses your camera field of view. The flight path of the meteor must be parallel or nearly so to the grooves of the grating for good dispersion. If it is across the grooves no spectrum will be obtained.