Only Amazon can invite you to join their program; so you can't just decide to join and be a part of it. However, everytime you write a review for a product on Amazon, and people say that it was a helpful review, then your Amazon rating will get better, and then they will invite you to become a Vine Voice product reviewer, and you will start getting products to keep and review.
If you do a search for amazon vine; it is mentioned in several different websites and blogs online, and some are stories from people who actually have become a vine reviewer.
Apparently, they will send you a list of products available to try and you can choose the one you want to test and review, according to what i read.
Another great place to test out products for writing reviews is called BzzAgent It might only be in the US, I am not sure. You would have to try and sign up.
Once you sign up for BzzAgent; you fill out a profile of what you are interested, and when they have products for you to test, they let you know.
You can see some of the products they are testing when you look on the webpage.
I have done this for a year or two, and tried some great products.
https://www.bzzagent.com