I have some small emerald cedars that need to be relocated between now and the middle of May. The weather is just starting to get nice, so I'm not sure if I need to relocate them now, before they "wake up", or if they'll be OK if I relocate them in a few weeks.
The Arborvitae 'Emerald Green' have already begun waking up. How big are they and how long have they been planted in their current spot? This is a very iffy project to do now. The only time to try this is in the late fall and only after preparing them to be moved.
Now would be a great time to start getting your arborvitaes ready to be moved. Hopefully they've been in the ground less than a year. Hopefully they are no more than 3' high. Otherwise, you are looking at a 50/50 chance of survival if you've prepared them to move, at least 6 months to a year or more before moving during the late fall.
The process is simply digging a 1' deep trench around the root ball the plant needs to support its biomass and the size you are able to move by yourself then filling that trench with straw. If your arborvitaes are 6' by 2' wide...you should go with a professional company. That root ball will have to be 3' in diameter and depending how closely you've planted these arborvitaes, means you're probably compromising roots for its neighbor.
Digging the trench and filling it with straw right now will ensure your plants are able to put out new feeder roots within the rootball. In this way the plant gets used to having its roots pruned, gets enough feeder roots in the root ball it will need now and when transplanted. One shock at a time.
If you were to dig up your plants now, move them to their new place, I'd give you maybe a 10% chance of success.