Is there a way to ask customers when they checkout where or how they found my website, for example Google, Yahoo, Bing search, from flyers, word of mouth ?
Building a custom form sounds like a good idea BUT.....
I've no idea on how to add this to my godaddy online store. The template and structure seems very rigid to me.
Google Analytics is typically the best way forward (and is free!). In Analytics you can see the traffic sources of all your traffic and sales, as well as useful information such as what devices visitors use, their geographical locations, the time they spent on the site, what pages they visited, how many times they visited before purchasing etc. You can do some great custom reports and create easy to interpret dashboards too.
Alternatively you could add a field into your checkout process to ask people. This may be a preferable option if much of your awareness and interest is driven from offline marketing.
Just shout if you need any help with Analytics or online marketing - www.herringbonemarketing.co.uk
Hope that helps!