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
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)) $.
