You can transplant rhododendrons any time the ground is not frozen if you are careful. However, it is best to root prune now and move in the fall. Root prune means to cut down with a shovel around the plants drip line where you are going to dig the root ball. Then in the fall you will have a nice dense root ball.

The down side with doing this is that they won't bloom until 2 years from now since next years flower buds are forming this summer.

If having flowers next summer is important to you then you can move when the weather is relatively cool, but be sure you can water this summer after moving and during any dry periods.

Here is a link that might be useful: How to grow rhododendrons and azaleas.

I would NOT transplant in this heat (expecting 95-100 F. for the next few days. Maybe cooler on Long Island, but still way too hot to transplant!! If we get into a stretch of cloudier, cooler weather, it's possible. But being that you are going from shade to sun, I really would wait until September (sun to shade transplanting is not AS problematic). I planted a new row of rhododendrons this past April. They face south but are really in shaded conditions (especially midday) due to a planting of maples. Well, they nearly all flowered and pushed rapid new growth. Wonder if light is the only factor in your lack of flowers. I have had mine flower in fairly shaded conditions in the past. PS. When you do transplant, I would improve the soil with some enriched peat moss and mulch around the plants. Then get them watered (if no rain) going through the autumn and into winter. Good luck!