So you want it to auto save to the cloud? You can always select where it saves to when you screenshot and download via FF or directly hit copy and you can paste it in your cloud folder through your OS files app.
Maybe linkwarden? The cloud hosted is paid, theres no free tier but theres free trial, its selfhostable if thats your thing, and theres few places that hosted it for free for the public to use. No e2e though so using the free public instance by random people might not be a good idea unless you really trust the hoster.
I tried it but the extension doesn’t have screenshot unfortunately , I ended up just saving locally and doing scheduled rclone copy to e2ee filehoster (Filen) , janky until i find proper e2ee solution , or when standard notes is included in proton subscriptions