Is it possible to make a good Tom Kha Kai without certain ingredients?

Simple answer is no, jenn. almost all of those ingreds are essential for the recipe.i don't think dried lemongrass has flavor; you should just trash it. just fyi, lemongrass lasts forever frozen-either whole or chopped or pureed.same w/ ginger. only fresh ginger works for savory asian cooking. it also keeps well frozen. i keep a knob frozen and it peels and slices easily in frozen state (Do not defrost first- it gets too mushy.)
lime leaves not as critical, but they also can be kept frozen, as well as the chiles, any chiles (Jalapenos, thai, any).

If the recipe appeals to you, then it's likely you will want to make more Thai dishes, and it would be worth your while to go to an Asian grocer to buy these things. You may also want to look through more recipes to see if you might need other ingredients for them, that you could buy at the same time. Some Asian stores carry products for a large variety of asian cuisines.one stop shopping is so helpful in our busy lives!