Early spring is probably best, and early autumn is probably second-best, but you might be able to successfully transplant them in-between if they can be regularly monitored for watering needs. (To be fair, any recently-planted/transplanted shrub should be regularly checked for water.) Azaleas have fairly shallow root systems, so as the soil surface starts to dry out, they may need more attention than a shrub with deeper roots. Dig as wide of a root ball as you can since the shrub's roots will have grown further outwards, away from the main stems, than they will have downwards. Try to move it on a cloudy, cooler day if possible, or temporarily shade the plant if you can for a few days after the move if the weather is sunny and warm.