The law applies to new and renewing transactions.  The session law in the bill makes clear that the new law applies to both new and renewing policies of insurance:

Sec. 3. Applicability. This act applies to new policies of insurance and renewed policies of insurance issued from and after June 30, 2023.

Furthermore, A.R.S. § 20-1510(A) refers to actions associated with a “policy of insurance” which includes new and renewal actions.  Therefore, an insurer may not decline to write, non-renew, cancel, or price new or renewing policies based solely on the possession of a certain breed of dog on or after that date.