How can I cook my dumplings without a bamboo steamer?
A couple thoughts... I haven't tried this, but I've heard of people doing it.

The microwave works relatively well to steam things. You could try "steaming" the dumplings in the microwave with a bit of water at the bottom of the dish or better yet, cover with a wet paper towel. You'll probably need to do some experimenting unfortunately. Under "ideal" circumstances, I'd probably prefer proper steaming, but this might work in a pinch. I imagine re-heating them would work well, but if cooking them works to your liking... no sense cooking them twice.

If you have a rice cooker, you could easily rig that up to "steam" as well. Just rig it up as if you were using a pot to steam it. Just make sure it isn't dry....