Is this the same as supplying a US customer directly (which they also do) where they don't - obviously - charge VAT?
Without seeing the contracts it ishard to be sure, but it sounds like the end customer is still the customer and your client is buying fulfilment services from Amazon...
I thought that it could be relevant if you had stock stored in the US.https://blog.taxjar.com/sales-tax-nexus-definition/
As a general rule, sales tax is due if the seller has a presence in the state. Whether having stock in a third party warehouse brings them into scope, I don't know.
This will be a commonplace problem and it's very likely that amazon (US) has some guidance for sellers. There are also seller forums where your client may be able to get help.
Maybe but do you really want to become an expert in US sales tax? Refer your client to an expert.