Thompson's group may act by homeomorphisms on the circle.
Has this action a fixed point?
If you mean Thompson's $F$, please, specify the action on the circle.
If you mean Thompson's group $[F,F]$, the answer is yes, as it acts on the interval.
I think you are talking about Thompson's group $T$, and its dynamics on the circle was well described by Ghys and Sergiescu in
The answer to your question is however rather trivial : the group contains every dyadic rotation, which act minimally on the circle. So every finite orbite must be dense, which is impossible.