An organization purchased a piece of property in July, would the property be exempt in the current year?

No. The property would be taxable to the January 1 owner.

Per KRS 132.220: Assessment dates – Listing – Owner – Liability – Exemptions, listing, annual review.

(1)(a): All taxable property and all interests in taxable property, unless otherwise specifically provided by law, shall be listed, assessed, and valued as of January 1 of each year.