In reply to alexm198:
Try asking some of the local guides. Googling Caucasus mountain guide and Elbrus mountain guide brings up a few websites.
climbing.ru is the Russian version of UKC but it’s all in Russian so not sure you could put a post in English on the forums.
Make sure you read up on all the necessary permits. I don’t know about Georgia but Russia is a bureaucratic nightmare or at least it used to be. Visa, invitation from a local hotel or similar and, depending on where you’re going, a border zone permit. Don’t know about peak permit fees. Assume the local authorities (police, border guards, etc) are corrupt and will use any flaws in the paperwork to extort you!
And keep away from areas with an unstable security situation... Chechnya, South Ossetia and Nagorno-Karabakh are the obvious ones. Abkhazia, North Ossetia, Ingushetia and Dagestan probably not great either. Fortunately Elbrus and Dykh Tau are further west.