In New Zealand, these are called “trundlers,” which I might pick over either of these. In terms of what we have, “shopping cart” has a straightforward, functional dignity, “shopping trolley” is a little more fun and makes more clear that you aren’t just shopping but shopping on wheels.
Jeremy’s winner: shopping trolley
I, too, would probably pick “trundlers” over either option, but “trolley” is more fun.
Beckie’s winner: shopping trolley