You can absolutely do that.
Option 1: You have the export of subscriber tokens with linked FCM account.
In case you haven't heard about FCM before there is a high chance your subscribers cannot be exported in an useable manner. Our customer support team can verify on your behalf to see if FCM account was set up and an export is possible.
In case you have FCM account setup, just send us your existing subscriber export with tokens and API key.
Option 2: In case your old push notification provider has doesn't provide an export, PushOwl can auto import your subscribers when they visit the site again. It means that the current subscribers don't need to give your store permission to send notifications again.
Pro tips: You can enable PushOwl on your store, and send few push notification campaigns from your old provider. So when the old subscribers click and visit again, our system will automatically issue a new push notification subscription, and transfer it to your PushOwl account. This happens behind the scene and your existing subscribers don't need to do an explicit action.