You can absolutely stir-fry green beans!
Make the sauce in advance: equal parts shoyu, sake, mirin, and sugar, stirred until sugar is dissolved. I usually do about 1 tsp each.
Heat your oil (sesame oil is best for flavor), add some garlic when the oil gets hot, stir just long enough to get fragrant, then add your green beans. Add some peppers if you like. Add your sauce gradually to coat the green beans. You may need to add a bit of water to steam them, but you shouldn't have any water left when you're done, just a nice bit of teriyaki glaze.
Serve over rice.
Edit: shoyu - soy sauce That's one of the words I grew up with in Japanese and never got comfortable with the English word.
Thank you!! This is very very helpful and sounds delicious! Definitely going to try this.

Also... probably stupid question... but where would I get sake and mirin?? Would those be in regular grocery stores?? Or would I have to go to a specialty store or something??