The answer is no, such a ``local slice'' does not exist in general even for algebraic actions.  Let $ \delta \in T_{k[x_{1},x_{2}]/k} $ be the derivation $ x_{1} \frac{\partial}{\partial x_{2}}+x_{2}\frac{\partial}{\partial x_{1}} $.  By the infinite series identities
\begin{align*}
  Let $ \widehat{\mathbb{G}_{a}} $ act on $ \operatorname{Spf}(k[[x_{1},x_{2}]]) $ via the action $ \beta $ which sends $ (t_{0},(a_{1},a_{2})) $ to $ (a_{1}b_{1}(t_{0})+a_{2}b_{2}(t_{0}),a_{2}b_{1}(t_{0})+a_{1}b_{2}(t_{0})) $.