Thompson's group may act by homeomorphisms on the circle.
Has this action a fixed point?
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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
Sur un groupe remarquable de difféomorphismes du cercle.
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.
