Hello, Tom Wolsky,
Thanks for using Apple Support Communities, it's great to have you here. We want to help you figure out what might be going on with your iPhone so that you can paste successfully. It sounds like you're copying paste in one app on your iPhone and then trying to paste that text on iCloud.com using Safari on that same iPhone? If our understanding is correct, then you should certainly be able to do this. We do want to note that some types of content can be copied but depending on the app, you may not be able to paste it. However, if the content you're working with is purely text only (and no images or other content) then you should be able to do that seamlessly.
Since that isn't working, we would recommend that you double check that your iPhone is running the latest version of iOS. This is because iOS updates can often resolve issues of unanticipated behavior like the one you're experiencing. This is how you can update easily: Update your iPhone or iPad - Apple Support
As for the issue itself, we would recommend making sure that iCloud is set up as anticipated on each of your devices to ensure that everything is able to sync over properly. Here is how to do that: Set up iCloud for Notes on all your devices - Apple Support
We hope that this information is helpful.
Have a great day!