[Vuejs]-How do I Notify the Client when a Payment Systems sends a Request to my API callback URL?

I think Websockets should work for your problem. As you already said you should persist the state of the payment in your database. When the Swish app calls your callback url you send an websocket request with the new state and needed information. The state is needed when the websocket connection is recreated due errors in the communication.

Some links which may be usefull:

• websockets/ws
• nathantsoi/vue-native-websocket
• sockjs/sockjs-client
• stomp-js/stompjs