The Legion, maybe. But the NCR is a diesel-punk state. I would much imagine them using motorcycles, modified bicycles, or other forms of wheeled single-passenger transport, compared to mutated horses or other animals (unless you want to invoke some crazy brahmin genetic experiment).
The Legion is more of a wild card because there is a possibility they could have assimilated mounts from the conquering of the 86 tribes. While it's unlikely they would have used mutated dogs for mounts (remember Ulysses's lines about how they broke the Hang-dogs), it's possible that another form of mutated animal mount (perhaps from one of their conquered states), could be utilized.
Overall, I think the NCR would not use animal mounts. They would use mechanical vehicles. The Legion has options though, potentially based on their regional roots in the US.