Yes, it could.
Based on your description, I have a feeling that stormstornadoes and hurricanes will be more frequent.
They have the nice (well, depends on the points of view) habit of carrying things around, and it can surely happen that some capybara is carried along the storm. The few which will survive the flight and land close to an island can settle up there, until the next storm.
It's a sort of Monte Carlo based traveling, but I think over a time span of 1000 years it will allow a decent spreading of the capybara around, reaching even Louisiana.
see also rain of animals
Under this hypothesis, a tornadic waterspout transports animals to relatively high altitudes, carrying them over large distances.