1. Understanding the problem: Alice starts at vertex $A$ of a regular hexagon $ABCDEF$ and moves either right or left with equal probability. We need to find the probability that after 35 moves, Alice is at either vertex $A$ or $C$.

2. Analyzing the movement: In a regular hexagon, moving right or left from any vertex will move Alice to an adjacent vertex. The vertices are cyclic, so moving 6 steps in one direction will bring Alice back to the starting vertex.

3. Distance and parity: To be at vertex $A$ after 35 moves, Alice must have made an equal number of moves to the right and left, which is impossible since 35 is odd. Therefore, Alice cannot be at vertex $A$ after 35 moves.

4. **Vertex $C$**: To be at vertex $C$ after 35 moves, Alice must have moved 3 steps away from $A$ (since $C$ is 3 steps away from $A$ in either direction). This means the net number of steps in one direction must be 3 more than the steps in the opposite direction. However, since 35 is odd, the net number of steps cannot be 3 (an odd number) because the total number of steps must be even to balance out the odd number of moves.

5. Conclusion: Since Alice cannot be at vertex $A$ or $C$ after 35 moves due to the parity of the number of moves, the probability is 0.

$$\boxed{0}$$