From an email from the City of Mesa -
Early voting for the Aug. 30 Arizona Primary Election began on Aug. 3 and ends on Aug. 26.Voters on the Permanent Early Voting List and voters wishing to receive an early ballot who are not registered with a declared political party must contact Maricopa County Elections at (602) 506-1511 to request which ballot they want to receive (Republican, Democrat, Green or non-partisan if it is a ballot for [non-partisan] candidates only). Voters, not registered with a party, who plan on going to the polls to vote can request the ballot they want upon arrival at the polling site.
While this email focused only on Maricopa County (home county of Mesa), the procedure is the same in other counties, except election questions should be directed to the relevant county's elections department.
Maricopa County's list of early voting sites is here.
Pima County's list of early voting sites is here.
Pinal County's early voting sites are the offices of the Pinal County Recorder; the list of those offices is here.
Yavapai County's early voting sites are the offices of the Pinal County Recorder; the list of those offices is here.
Yuma County - early ballots can be dropped off at the county recorder's office here.
Santa Cruz County - contact the Elections Department at 520-375-7808 for early voting information. The full list of voting centers for the election is here. Note: Santa Cruz County's website is vague, but I do *not* believe these are also early voting locations.
Coconino County's list of early voting sites can be downloaded here.
For the other counties, please contact their elections departments for the relevant early voting information.