I am a piano teacher and I wouldn't take a 5 year old either, sorry. I sometimes take 6 year olds but the parent has to be clear about what they're getting for their money. A 6 year old will generally make progress much more slowly than, say, an 8 year old, and you will find that by 11, a child who started at 6 is often very little further ahead than one who started at 8+. If you are happy to pay just for your child to have fun - and can accept it will to be a much slower, more play-based experience, then fine.
Having said that, I did start my 10 year old just after her 6th birthday, mainly because she'd been nagging me for about 2 years. But then, I wasn't paying for the lessons .
And true to form, and despite the fact she is now a very capable and competent little pianist, her best friend who started with me at 8 is only about 2 terms behind her if you're looking at grades. It wasn't a mistake to start her so early because she loved it and still does, but her progress was very slow for the first year or two. I knew that would be the case and we just had fun, but I worry that parents of such young children will expect too much too soon and put them off for life.
As to how to find someone, I would go by recommendation if at all possible, ask as many people as you can think of if they know of anyone. Also, a teacher who is fairly full is more likely to take recommended students. Failing that, get a list from the music shop, phone a few, not just one, and see who you click with and like the sound of.