If it’s your account, then you’re the seller of record to the buyers. This means for each sale you’d have a few transactions:

1. Buyer pays you
2. Amazon takes fee
3. You pay the “family business” for the item (plus postage?)
4. You keep the remainder

Number 1 is income, Numbers 2 and 3 are expenses and number 4 if your profit.

If you’re based in the UK, then HMRC don’t require you to register as self-employed until you have (or expect to have) £1,000 (Edit: gross income) in a tax year. It would be safest to register anyway as Amazon require proof of registration with HMRC if you’re operating as a business (e.g. buying stock to resell, or acting as a middle man).

Note also Amazon’s policies about dropshipping: https://sellercentral-europe.amazon.com/gp/help/help.html?itemID=G201808410&ref=efph_G201808410_cont_521

If it is their seller account and you’re operating it for them, then the following would apply:

1. Customer pays them
2. Amazon takes fee
3. They ship the item
4. Remainder is their profit
5. They pay you whatever fee you’ve agreed

You don’t need to worry about items 1-4, and only need to account for item 5.