Stopped receiving emails and need more storage

I’d first check which storage is full. Go to Settings > your name > iCloud.

If you use an @icloud.com email address and iCloud is full, that can stop new messages from arriving. Tap Manage Plan and choose a larger iCloud+ plan, or free up some iCloud space.

If your email is Gmail, Yahoo, Outlook, or another provider, buying iCloud storage won’t increase that mailbox. You’ll need to check the storage limit with that email provider instead.