Yes. They transplant very easily. In fact, it was my uncle's way of 'fixing' azaleas. If an azalea wasn't doing well, he would dig it up and replant it (in the same place).
You don't say where this azalea is located (country, state, city...) or what kind of native soil you have. But, you will need to add a goodly amount of milled peat. I lived in river bottom (in California). I used half native soil (which was humus) and half milled peat in an azalea planting hole.
If your soil isn't the best, replace it with half garden soil and half milled peat. Luckily, azalea have pretty small rooting systems so a hole that is between 5 and 7 gallons is more than plenty. They are very shallow rooted so the hole needs to be wider than deep (maybe 12 inches deep and 18 inches wide). Essentially, you are planting in a pot without sides under ground. They stay in whatever space you give them and use whatever nutrients are available.
Make sure the peat and soil is completely damp (no dry spots) before planting your azalea. Otherwise, you risk burning the roots off.
