It is hard to tell what is there but the first thing that strikes me is that the soil looks dry and is not mulched. Azaleas like well draining, acidic soils that are as evenly moist as you can make them and with 2-4" of organic mulch. Dappled sun and a few hours of morning sun is fine but it is hard to guess how dense the shade gets from the picture. Full but bright shade may be ok for some varieties. The best time to transplant is when the plant is dormant and the soil is not frozen so, in the Fall or early Spring. The roots are usually shallow, tiny, fibrous and around 4" (or thereabouts) deep only so be careful and try to move as much of those roots as you can. You could try transplanting in the Fall but only if frost heave is not a problem where soil freezing and thawing cycles occur. Raised beds or pots are alternatives in some places.

I assume that is a deciduous azalea and they prefer a lot of sun! It is not surprising yours is not doing well under the current conditions.

It will be climate/location dependent, but I would not have any issues transplanting this whenever I was able. IME, rhodies and their azalea cousins are very amenable to transplanting, although I would avoid if flowering or in the middle of summer heat. And neither of those appear to be a factor in this case :-) My only concern would be how the azalea roots might be entangled with those of the nearby tree.

It is not thriving so what do you have to lose? The ones I moved had roots deeper than 4" but they were probably older than that one.

I'd have no problem relocating it, but I'd plan ahead. Get the new hole dug first so it doesn't have to lay around waiting. Lay it on a tarp and drag it to its new location instead of carrying it around by the stems. Remember that cut roots will heal more cleanly than torn roots- no yanking or pulling. And if it shows stress be ready to rig up some temporary shade using sheets and clothespins, stakes, chairs, or anything else you can utilize. And do deep watering when it needs it- a walk-by with a hose just doesn't cut it as far as seeing that the water actually penetrates.