I am having the same issue. I updated my site with a new woo theme (storefront) and updated woocommerce to the latest version. When I am logged in, I can add items to cart, but when not logged in, nothing goes into cart. I did notice that two tables do not exist:
woocommerce_sessions Table does not exist
woocommerce_api_keys Table does not exist
Could this have something to do with the problem, and if so, what do I do to fix it? Please note: I am not a technical expert, and need step-by-step instructions how to fix.
Thank you!