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.

I would not use it. SAJ14SAJ's answer goes into the physical properties of glass, but I would have a wholly different problem: it would not be steaming.

For steaming, you want the 100?C hot steam to rise to the food and condense on it. But if you put the food in a bowl (which is closed from the bottom), this can't happen. You will have the vegetables in a closed bowl in a hot environment, which is essentially the same as baking, not steaming. You can do this if you want to, but the oven will be probably less fuss for the same result.

If you want to steam something, you want the steaming container to have holes so the steam can rise through them and through the food pieces. You can buy cheap metal steamers working on a diaphragm principle, so that they fit different pot shapes. Bamboo steamers also tend to be cheap in ethnic stores. But if you really don't want to buy the actual tool for the job, your best substitution is a metal sieve or colander. Choose a pot where the sieve/colander can be suspended (or placed on a cylinder, e.g. upturned mug) some centimeters above the bottom, put the water on the bottom, and the food in the sieve. Cover the pot and steam away.

Since I use a rice cooker as a steamer, my solution to the thermal shock problem is quite simple. I set the rice cooker to "Warm" rather than "Cook" mode for a few minutes. Only then do I switch the cooker to full "Cook" mode. My rice cooker also has a "Congee" mode for cooking rice gruel or chocolate porridge. So I sometimes also use that for the added convenience of a single slow cook setting.