Can you maneuver a pickup truck near enough to pull out the stumps and roots? That's what I did when I removed my old juniper. It came out pretty easily, leaving only a couple of roots that I used an ax to remove.
The azalea should do fine there, even if deeper roots remain. They're quite shallow-rooted.
Azaleas are often forced to bloom right around Mother's Day. Those are the ones usually found in florists shops and grocery stores, intended to be gifts. They can survive when planted outdoors, but they often don't perform very well for the first 2 or 3 years, and they need some babying to get over the forcing.
Since you bought yours at a regular plant nursery, they're probably ordinary outdoor azaleas.
I don't know where you're located. Make sure the soil is acidic for the azaleas, as they won't do at all well without that. Azaleas can take more sun than rhodies. I have had them grow very well in places where they received full sun from dawn until about 2 or 3 p.m. Being shallow-rooted, they aren't all that hard to transplant, if you decide you want to move them later on. They'll need daily watering for the first few weeks, so the roots can get established. They'll need regular watering through the summer, and into the fall. If the ground freezes where you live, it's a good idea to water them well before a freeze, because the roots can't absorb water from frozen soil.
