There's one practice that I have seen that implies that a minyan is not required for Kabbalat Shabbat.
If I recall correctly, our shul had exactly 10 people for the minyan. An avel (mourner) came to shul and davened mincha Fri. evening. Just prior to Kabbalat Shabbat, the rabbi asked him to leave the shul and he returned just pior to reciting Mizmor Shir Leyom Hashabbat (Psalm 92.) This is because Kabbalat Shabbat is considered a "separate service", and technically, it is "not yet Shabbat " (We have a M.Y. question explaining this - I have to locate it.) so that person is still "in mourning display mode." Once Shabbat, is "official", which occurs with reciting Psalm 92, the mourner accepts Shabbat and he may not display public mourning.
The point is, that this action indicates that a minyan is not needed to say Kabbalat Shabbat.
There is a separate angle as to whether the recitation of Psalm 92 is the only method to indicate Shabbat acceptance, and even when used, does an individual reciting that Psalm accept Shabbat, or is it only for a tzibbur. I will try to edit that argument in, later.