Sure; just note you won't be proving fermented foods lessen social anxiety -- at most you can conclude that consuming kimchi does it.
It's pretty good so you're not at loss anyways! I find sauerkraut enjoyable too, vegetables are too bland for me, a little bit of this stuff is a sauce replacement for me.
Unless you know a vegetarian, you may have trouble finding gel capsules sufficient for a decent dose of kimchi... (Also, now you have to refrigerate all your capsules.) Easier to just use some probiotic powder stable at room-temperature unless there's some compelling reason to expect them to be ineffective.
Only if you can find some non-fermented placebo kimchi that looks and tastes just the same, otherwise it ruins the blindness of the experiment. Hmm, Placebo Kimchi sounds like a good name for a band :)