Hi
Had the same on the coffee shop once. Not WooCommerce, the host's page cache was handing out a saved copy of the cart page.
You say no cache plugin, but SiteGround caches on the server anyway, I'm pretty sure it is on by default and that is why it works on your laptop, you are logged in and get a fresh page while a buyer gets the stored copy with an empty cart.
Check Speed Optimizer or Site Tools. Dynamic Cache on, but you need to have /cart, /checkout and /my-account in the exclude list. Easy to miss.
If those are missing, add them!
Then purge the cache and let one buyer try again.