Welcome and great to see another Laravel and Vue developer here!
Is the app going to initially load its data via the API when it has a network connection? It kinda sounds like you are going towards a offline-first approach. One option is Ionic Storage. Here is a link to how I implemented it in Vue and Vuex - How to use Ionic-Storage in Vue 3.0? - #9 by twestrick.
In regards to saving updates offline and syncing them once there is a network connection, I think there are at least two ways to approach it. The simplest option in my mind would be to check if there is a network connection and if there isn’t, save the request in local storage. Once internet is back, replay the requests.
A more complicated/robust approach is to build out a similar data structure locally in SQLite and then do a full sync between the local data and backend data.
Things to think about are handling conflicts between the backend data and app data when the backend gets updated before the app syncs.
Here is another discussion that might be helpful. I think I’ve seen several others too on these forums - Offline-first app development (local storage + server sync)