Yes, you can use a handheld steam cleaner to kill bed bugs. These products can heat up the surface area of the mattress enough to kill bed bugs in all stages of the life cycle.
In order to kill the bed bugs and prevent damaging your mattress, you want the surface temperature on the mattress to get between 160- and 180-degrees Fahrenheit. In order to ensure the surface gets hot enough, without getting too hot, try using an infrared thermometer to measure the temperature immediately after passing your steam cleaner over a section.
If you find that the temperature is too hot, you should move the steamer more quickly. If you find that the temperature is not hot enough, try moving the steamer more slowly to let it get each section hot enough to kill the bugs.