Of course, the fact that it is the accusers who want the FBI to investigate, and the Democrats, and not Kavanaugh and the GOP is pretty telling in and of itself.
So looking into it, they can direct the FBI to do another background check digging into the claims, though from what I've read, it seems there were more witnesses and the like to be able to check with for Clarence Thomas than with the current first allegation.
Regarding who is asking though, my suspicion is that the Democrats are only wanting it investigated, so that it will slow the process enough he can't be confirmed until after the elections, and the Republicans obviously taking the opposite tack, not wanting it investigated so it doesn't slow the process down.
So neither side looking that good, but I don't think it is telling that the Democrats want it investigated, as I feel they only want it investigated for political reasons.
For the accusers, it could be either they are pretty confident, or that they for some reason want to slow the process down as well, and for Kavanaugh, it is either he is guilty so doesn't it investigated, or is innocent and doesn't want it slowed down / risk not being a Supreme Court Judge because of an investigation.
I don't know, obviously, where the truth lies, and agree that ideally we would have an investigation to clear it up - but only potential issue I see is that an investigation could clear Kavanuagh, but cause enough time to elapse that he doesn't end up getting the position on the court, and thus whether there is a way to mitigate this outcome.
On the flipside, and I don't know what the laws are, is it possible for him to be placed on the court, an investigation to still take place, and him be thrown off the court later if found he was guilty as such?