I converted to sconces to plug in lamps with on/off switches as there was no readily accessible wiring for the area. The back of the sconce has a bare copper ground. I bought wire that has only a hot and neutral. If I mount these sconces on the wall, do I really need to connect the ground? Do I risk an apartment fire if I don't? And if so, why is it that all other lamps have only two plugs?

Solutions?

No,a UL Listed lamp does not need to be grounded. As long as it uses UL Listed parts, such as a polarized cord and plug, socket, which should be since it came with the fixture, etc.

I don't see UL getting all upset over homemade lamps.

The cord may need to be grommeted, or have some sort of bushing to prevent chaffing the cord.

Polarized means the white terminal on the socket receives the ribbed conductor on the cord ( look close you will see ribs or dashes) and uses the wider blade on the plug.

I converted to sconces to plug in lamps with on/off switches as there was no readily accessible wiring for the area. Lost me! Did you convert from sconces to lamps??
The back of the sconce has a bare copper ground. I bought wire that has only a hot and neutral. If I mount these sconces on the wall, do I really need to connect the ground? Yes! But your first question has me confused? Are the sconces made of conductive parts?
Do I risk an apartment fire if I don't? And if so, why is it that all other lamps have only two plugs? TK answered this.