Hi,
I checked on my end and I have the same issue with Chrome and Firefox on my desktop on your website.
The fact that when, I access the checkout, I get redirected to the listing of products indicates that the cart is empty.
However, when I click on add to cart, I get a successful message.
Also, each time I click on an "add to cart" button, if I check the response from the server with my browser's developer tools, the id of the cart returned is incremented.
This means that the cart is lost on each page of your website, as if the session data wouldn't be saved at the end of each page.
Through the years, we had people with a similar issue, but usually it is related to a server problem.
For example:
www.hikashop.com/forum/checkout/878972-s...the-cart.html#212646
So try changing the "session handler" setting of the Joomla configuration and check whether you've change anything regarding session, or security, on your website / server.