Backup happens on its own
There's nothing to switch on and no backup button. Having an account is the backup. Whenever you're online, what you've captured is saved to it.
This is what an account is for. Without one, your contacts exist only on that phone, and there is nothing to restore from if it's lost.
Get your contacts back on a new phone
On a fresh install the Login button is at the bottom of the opening screen, easy to miss under the sign-up and subscription prompts.
If you're past that, use the three-line menu at the top left and choose Sign up / Login.
Wait some minutes and the contacts will start returning in your device
Moving your subscription too
Restoring your contacts and restoring your subscription are two different things, and people routinely do the first and wonder why they're still limited.
Your contacts follow your Covve account. Your subscription follows the app store account you bought it with. To bring the subscription across, install Covve using that same App Store or Google Play account, open the upgrade screen, and tap Restore purchases.
One account, one person.
A Covve account and its license are for a single user. Sharing sign-in details across a team isn't how team access works. For that, everyone needs their own license and the leads are shared properly. See Set up or join a team.
FAQ
Do I need to do anything to back up?
No. Having an account is the backup.
Will restoring overwrite what's already on the phone?
Restoring brings back what's in your account. If you captured contacts on the new phone before signing in, check everything is present afterwards.
Does this work without an account?
No. Your contacts stay on that one phone and there's nothing to restore from.