Some card issuers don't require it done via a phone if that's important for you.
And some (looking at you, DBS) toggle seemingly at random between requiring it via SMS or via their mobile app.
The card issuer can send you a one time link to a web portal by email or using an iframe, where you log in and confirm the transaction.
Presumably if you're initiating an online purchase, you have access to WiFi?
Some card issuers don't require it done via a phone if that's important for you.