Libraries are a great source for finding free books to review. There are libraries with great budgets and donors, so they are always getting in new releases. There are also smaller libraries that get few new acquisitions, but they often have a good back stock of older books that you may still be able to review on your blog.
If you live near a college or a university, stop in and find out about getting a library card from them. I did this years ago at Kutztown University and I borrow hundreds of books from them every year.
Free Kindle Books
When writers publish their books on Kindle, they have the option to join the KDP program and set free days for their books. These free books can be found by going to the Kindle Books section, clicking on Best Sellers, and then clicking on Top 100 Free. You can check out the Top 100 Free for each section.
Used Book Stores
I stop at my local Goodwill shop about once a week. They have a large area for books, priced at 25 cents each. I have found some wonderful gems, including brand new books. The one time I went there, they had over 20 new books about writing, so I bought them all for around $6.
Used book stores carry a lot of new books and they cost very little to buy. Older books are also a great source of inspiration and I use these books for research and for generating new articles and blogs.
Build Up a Stock of Published Reviews Before Contacting Publishers
I began reviewing books professionally for publishers and publicity groups over a decade ago. I didn't just jump into this opportunity. In fact, it found me when a publicist contacted me about reviewing a book on my website. I had already had a large amount of book reviews on my site. After I reviewed the book, I created a book review request letter to send out to publishers and publicists. The letter provided links to the books I had already reviewed on my site and stated the theme / niche of my website.
Breaking into professional book reviewing is fun work, especially if you love, love, love books.