Send them an email asking for a quote. For a small business like this, I'd expect they can probably ship it to you but the complexities of quoting shipping internationally means they don't have it on the site. Yes, shipping APIs are readily available, but implementing them costs more than the shoestring budget they likely have for the site and would also require integration with their inventory software. I'd be willing to gamble that they are likely using a homegrown WMS that is a combination of excel sheets and several layers of scripts from former and current employees who can write code but aren't professional developers.

I've consulted for similar size businesses, and shipping integration was always the reason to not support international shipments. Send them an email, you'll almost certainly hear back from a real person and if you let them know what you want to order they can probably make it happen.