I'm an indie author and my customers are... readers.
I engage with them by running contests. For the launch of my last book I offered readers a free Kindle and also $50 of Amazon vouchers and my book. This was a great way of growing my email list, promoting the book and rewarding readers.
If you're running a contest I recommend using the WordPress plugin Kingsumo giveaways. I also suggest picking a giveaway that's relevant to your product or service. In other words, don't just give away a iPad.
When I'm not launching books, I try to respond to every email I get from customers or readers. I could outsource this but that strikes me as defeating the point of finding out what readers want and who they are.