I was able to replicate the issue you describe where when signed into My Verizon on my desktop, editing a Cloud contact does, in fact, remove the picture associated. I also found that, by clicking on the picture/(what was left in place of the picture) you can update the picture from your files on your PC.
Where the Verizon Cloud gets the pictures associated with Contacts has been a mystery, discussed previously related to the PC Desktop Message+ application and where it gets its Contacts. I'm not backing up Photos from my phone to my Cloud (free Cloud service only), so there are no pictures there, but my Cloud Contacts still have pictures from somewhere.
On my phone (Droid Turbo 2), I can add a picture to a contact, use the Verizon Cloud app and backup my phones Contacts. The picture change does NOT get to the Cloud Contact entry.
Maybe someone from Verizon can pass this on to the group that handles the Cloud storage to help us understand why editing a contact removes the picture, where these pictures are coming from and why pictures associated with our phone contacts are not getting to the Cloud when backed up.
"On my phone (Droid Turbo 2), I can add a picture to a contact, use the Verizon Cloud app and backup my phones Contacts. The picture change does NOT get to the Cloud Contact entry."
Update: after logging out of the Verizon Cloud on my desktop for awhile and going back in, the picture update I made on my phone to a contact DOES show up in the Cloud contact. I don't think the change made using the desktop login to the Verizon Cloud would get down to the phone contacts, since the phone's Cloud backup is one way.