They can. It would depend on how hard it was raining. You will need to make sure the controls are covered.
It rained while I was there in early January, so I purchased a Disney poncho and wore it spread over me and the ECV. Then had a second one to drape over it when I wasn't on the ECV.
Just a side note: You also have to use extra care when driving it through the parks in the rain because with the poncho hoods on people pay even less attention to where they are walking. They are less apt to look before they veer off the path.