No in general (in dimensions ≥ 3).

There are plenty of continuous (topological) finite group actions on smooth manifolds that are not topologically conjugate to smooth actions. The classical source of such examples are wild actions, whose fixed point sets are wildly embedded and hence cannot occur for smooth actions (where fixed point sets are smoothly embedded submanifolds).

Concrete mechanism:

• In dimension 3, there is Bing’s involution of S^3: a Z/2-action by homeomorphisms whose fixed point set is a wild Cantor set (hence the action is not locally linear and not smoothable). This involution can be arranged to be supported in a 3-ball and equal to the identity near the boundary. See e.g. Montgomery–Zippin, Topological Transformation Groups, for discussion of such wild involutions; Bing’s original constructions are classical.

• Consequently, on any smooth closed 3-manifold M, choose an embedded 3-ball and implant Bing’s involution inside it, extending by the identity outside. This gives an effective continuous Z/2-action on M which is not conjugate to any smooth action (its fixed set is wild).

• In higher dimensions n ≥ 4, take the product of the 3-dimensional example with the identity on D^{n-3} to get an involution supported in an n-ball. Again, you can implant it in any smooth n-manifold.

Therefore, the absence of smooth finite group actions does not imply the absence of continuous finite group actions: one can always create a wild continuous action supported in a ball.

To complete the counterexample, pick a smooth manifold that indeed has no nontrivial smooth finite group actions. For example, there are closed hyperbolic 3-manifolds with trivial isometry group. By a theorem of Kojima (and related Meeks–Scott rigidity results), any finite group of diffeomorphisms of a closed hyperbolic 3-manifold is conjugate to a group of isometries of the hyperbolic metric; hence if the isometry group is trivial, there are no nontrivial smooth finite group actions. Yet, as explained above, such a manifold still admits a nontrivial continuous Z/2-action supported in a ball.

Remarks by dimension:

• In dimension 2 (and 1), every finite group action by homeomorphisms is topologically conjugate to a smooth action (Kerékjártó–Eilenberg for surfaces), so in dimension 2 the implication does hold.
• Starting in dimension 3, wild (non–locally linear) actions exist, and the implication fails.