Are you talking about your Verizon Cloud service? You should be able to log into MyVerizon.com and access your Verizon Cloud by selecting it in the Account dropdown list. You can also see how to get the desktop app at https://www.verizonwireless.com/support/knowledge-base-128726/
It also has assistance for downloading content to your computer at https://www.verizonwireless.com/support/knowledge-base-200364/
Downloading your pictures can be incredibly useful. We want to make sure you can enjoy them on any device you need. Pictures can absolutely be downloaded. You can find the steps at https://www.verizonwireless.com/support/knowledge-base-79147/ . Does this help? What is the reason you're wanting to download the pictures?
The point of having data in the cloud is that you can access the data from anywhere.
The answer to your question is most likely yes, however there a MANY cloud services(A few of the most recognizeable services would be iCloud, Google Photos, Google Drive, Microsoft One Drive, Box). Each service would have its own method of accessing the data. Personally, I stay away from carrier specific cloud services since you lose access to the data if you choose to switch carriers and forget to download everything before you do so. You would be able to access the ones listed above from ANY device on ANY carrier or ANY ISP. If you are using Verizon's cloud service, you would only have access to the data as long as you were a customer. If you choose to switch carriers and forgot to download your data or back it up somewhere else, you would lose it.
If you could be a bit more specific on which cloud service, possibly more information could be provided to help you access your photos.
You won't lose your cloud content with all carrier based cloud services. Verizon Cloud will allow you to keep your service if you ever switch carriers, why you would do that is beyond me as long as you update your cloud profile with a verified email address so we can contact you with instructions for reactivating your cloud service on the new carrier (why would you do that??? )