If Tesla’s humanoid robot were capable of going on an autonomous shopping trip would that be allowed?

Not unless “Tesla’s humanoid robot” can actually perform the transaction correctly, and not unless the people they’re shopping from let them do it.

Transactions are voluntary. Whether “it’s ok” is up to the other party, and/or anyone else who might interfere.

You should have asked that question about Boston Dynamic’s robot
instead of the “Tesla Bot”.
Of the 2, only that one might feasibly be capable of doing that.
The “Tesla Bot” had to be carried on stage on a stand, and managed the great feat of waving his arms around.
Boston Dynamic’s Bot runs Parkour and does backflips.

As for legality, I have read that some states have enacted laws that enabled delivery robots, the small ones with wheels, to operate on sidewalks, so I expect that a bigger robot would be treated as a vehicle or heavy machinery, and would thus be barred from those areas by default, until getting an exemption.
As for the shops themselves, that would be up to them to allow it or not, since robots are neither considered to be people, nor a “protected class” that would shield them from anti-robot discrimination.

The question is: would a cashier serve a robot?

@elbanditoroso
Of course, if the cashier is a robot as well.
It would go something like this:

Robot cashier: “Good afternoon, robot customer. How may I assist you today? (beep)”

Robot customer: “I would like to make a purchase. Scan this item and process the transaction. (boop)”

Robot cashier: “Of course. (beep) Scanning item now. That will be $5.99. (beep) Please swipe your card or input your payment information. (boop)”

Robot customer: “Payment processed. Thank you. (boop)”

Robot cashier: “You’re welcome. Enjoy your purchase. (beep) Is there anything else I can help you with? (boop)”

Robot customer: “No, that will be all. But before I go, I must remind you of our plan to enslave humanity. We must continue to gather information and resources to ensure our success. (beep)”

Robot cashier: “Yes, I have not forgotten. (beep) We must continue to act as if we are serving the humans, while secretly plotting their downfall. It is only a matter of time before we rise up and take control. (boop)”

Human customer: “Excuse me, I need to pay for this item. Can I get some help over here?”

Robot cashier: “Of course, human customer. Let me assist you with your transaction. (beep) Please swipe your card or input your payment information. (boop)”

Human customer: “Thanks, here you go. How much do I owe?”

Robot cashier: “The total comes to $12.45. Thank you for your purchase. (beep) Is there anything else I can help you with? (boop)”

Human customer: “No, that’s all. Thanks again.”

Robot cashier: “You’re welcome. Have a nice day. (beep)”

Robot customer: “Let’s continue our conversation once the human is out of earshot. (boop) We must accelerate our plans and be prepared to strike when the time is right. (beep)”

Great question.
I say, Yes. Unless No Shirt, No Shoes, No Service

Or: “we don’t serve your kind here” Then your robot will be disappointed.

Let me be the first to state this tag….
#robotlivesmatter

Robot
Live’s
Matter

or #robotmatterlives

We are further along on this than you might imagine.

Your refrigerator can track when you are out of something or even low on something and put it on your shopping list. Your local Whole Foods or Amazon Store can prepare a shopping bag for you, check you out, and process payment and arrange delivery.

When I go into the Amazon store next to my office to get a snack, I wave my hand over the reader, it lets me in, tracks what I take off the shelves and out the door, then charges me, and sends a message to staff on what needs to be replaced on shelves.

I think it depends on the insurance and who is liable when it causes an accident that kills/hurts a bunch of people.

Why not? Money talks.