Times have changed, its not 20 or 30 years ago where people didn't have access to the internet. If a business chooses to advertise an email contact then I think it is perfectly reasonable for any correspondence to be delivered through there. In fact, I prefer email to phone calls or face to face communication, because with those you have no "paper trail" that you can rely on later. By sending and receiving emails you can use the proof to clear up misunderstandings later on.
In my experience, most correspondence between an employer and myself was made over email. Job applications, pay receipts, requests to work different days. When I did give my notice to leave they said that's fine but to send an email with the information. Most aspects of my life is addressed by email communication; applications to university, my housing application, contract and acceptance was all by email. Any communication with my university is preferably by email, you call up or go in and they ask you to email instead, and this is a pretty standard occurrence in the modern world, it's basically instant post. A business is not obligated to use email, but if they don't want they should not display it on a public website.
Although, I always like to add at the end of emails "please reply to confirm this" or something like that, and if I don't get a reply I will call to follow up, and at the end of the phone conversation I ask the to confirm the conversation with a response to the email or something.
Saying all this, it does not sound that you had a formal employment anyway. You were just helping out with horses in return for a discount, without a contract or anything. It's good that you gave notice, but I don't think the owner should act like she has been.