Can I use normal household glass items to steam cook inside pots?
Note:  I am assuming this glass is kitchenware, like mixing bowls or measuring cups, not service ware like drinking glasses or teacups.

Glass melts at about 1500 F / 800 C.  There is no danger of melting the glass in any type of steamer, or realistically with any equipment you may have at home.

The real issue is thermal shock:  very rapidly cooling glass from a hot temperature to a cold temperature can cause it to crack or shatter.  You want to avoid doing this.  The same thing can happen in reverse from cold to hot, which is why you would never use glass cookware on an open flame or burner element of your cooktop (hob to our British friends).

To prevent thermal shock, even in a closed pot used as an improvised steamer, do not place the glassware directly on the bottom of the pot.  It should be suspended on a rack or otherwise raised from the bottom.  If no other solution presents itself, aluminum foil loosly crushed into a ring will do the job (loosly, because aluminum is a good conductor or heat, but air is not).

Steaming is fairly gentle in terms of heating, and if you let it cool on the counter, you should be fine.  Note that I would not do this with any delicate or thin glass.  Think about it this way:  if you would use the glass in the microwave, the steamer should be fine as well.

Of course, you can always steam on a lettuce leaf or similar, which is very traditional, or use a metal steamer basket as well.

Please also see Rumtscho's answer for more optimal alternatives.