You can download a version -
instructions
If you don't have your vaccine letters (I didn't), you can get logged in if you know one of your vaccine dates and then use
forgot username
Then you can complete the registration and download a copy.
If you don't have your letters or know any of the dates, you'll likely need to phone the helpline
0808 196 8565.
Not sure if they'd tell you anything to let you download, or you'll need to wait up to 14 days for a paper copy - if anyone does it, comment and I'll update this.