You can transplant any time you want as long as you do so correctly, site them well, and observe the plants afterward and act quickly if they need temporary shade or a change in watering. If you yank them out and stick them somewhere in blazing sun and forget about them, maybe not.
For me it's more about the care with which it is done than the month of the year we are in.

Have they bloomed yet? If not, do you care of they do or is getting them relocated more important?

Their shallow rooting habit makes azaleas reasonably easy to move - best done when the ground is not frozen (and that hardly applies to you:)) or they are not in active growth. The period of active growth is the stage immediately after blooming when the new foliage buds are unfurling and opening. Timing the move before, or even during blooming is OK...Be careful not to plant any deeper than originally growing, and mulch the root zone to conserve moisture, cool the roots.

Dear jpapahall Florida,
Spring came early in the Southeast this year and we're experiencing drought. You would, therefore, be wise to wait until late fall/winter to transplant your azaleas. The azaleas will be much better off if they are transplanted while they are fully dormant. That will make things a lot easier on you, too. Less watering will be involved, and the azaleas will have a chance to develop their root systems in their new location while they are dormant. It is a little too late to transplant the azaleas this spring, based on our regional growing conditions.