When's the best time to plant azaleas?

It would depend upon where you are wanting to grow them. Please let us know where you are located or what the zone is.

I live in North Carolina.

Our climate here on the Gulf Coast is quite different from yours. Perhaps someone who lives nearer can be of more help. In general, here, we plant shrubs and trees most successfully in the fall.

Now would be a really good time to set out your azaleas in NC.
Y'all have a lot of native varieties up there...

Usually... The azaleas are planted in the Spring, because they're in bloom... But it isn't the best time... Unless you are in the mountains... When anytime is a good time... Very easy to garden when you live in a rain forest... But if you're down on the flat.. Less easy...

Fall is much better than Spring for azalea and rhododendrons. But everyone loves to buy things in bloom in Spring. It is however, starting to get a bit late. I am in NW Missouri and try not to plant after ~ October 15. I have planted as late as November 10th. BUT .... plants need to have time for their roots to get settled before they shut down for winter. If not, the plant has a higher probability of perishing.

It would be helpful to know the original poster's city and zone.

It will depend on whether the azalea is evergreen or deciduous, and where you are in North Carolina. If your ground freezes, fall planted evergreens can have a hard time because roots won't have sufficient time to recuperate to supply the overwintering leaves with their demand for water in the drier air of winter.

North Carolina is a big place and varied growing conditions. I think the USDA hardiness zones are from 6a to 8b I agree it would depend on exactly what climate zone, growing conditions and which type of Azalea. There are (I think) about 16 native Azaleas to choose from, each has somewhat different requirements. Maybe start by selecting the varieties which would like to grow in your zone and in your garden micro-climate; that could narrow down the answers.