Looking at the page, the button isn’t hidden in the sense that if you supplied the correct input it would display. The add to cart form hasn’t been generated and inserted into the code.
In the theme demo’s name your price product (http://demo.organizedthemes.com/grassroots/product/give-any-amount/) I can duplicate that by removing the regular price from the item. In the attached screenshot it’s the 50. When I take that out the code ends up just like it does on your page there. I’ve run into that before with customers so I just wanted to make sure that isn’t the case here.
If that doesn’t do it, try changing the product temporarily so that it isn’t name your own price. See if that brings back up the button. Since the button isn’t in the code on the front end, we know it has to be something on the backend that’s preventing it from being generated. Let me know what you turn up.
Thanks for doing that Henry. It looks like there is a conflict with the “WooCommerce Price by Country” extension. If I were to wager a guess, it’s likely that both this extension and the name your own price one are trying to modify the same functions in WooCommerce and it’s breaking the add to cart button.
I left the “WooCommerce Price by Country” extension turned off so the donation button would come back up again. I wish that they would play nicely together, but that doesn’t look like it is the case at the moment.