How do I handle a 1099-NEC that I received from Amazon for participating in their Amazon Vine program. This is not business income, it's just a hobby.

It doesn't matter whether you sell the items or not. Writing reviews for compensation is a business. The fact that the compensation was in-kind (products) rather than money doesn't change that. The value of the products that you received, as shown on the 1099-NEC, is business income.

To enter Hobby income and expenses -

Click on Federal Taxes (Personal using Home and Business)
Click on Wages and Income (Personal Income using Home and Business)
Click on I'll choose what I work on (if shown)
Scroll down to Less Common Income
On Miscellaneous Income, 1099-A, 1099-C, click the start or update button

On Hobby Income and Expenses, click on the start or update button

Or enter hobby expenses in the Search box located in the upper right of the program screen. Click on Jump to hobby expenses