vue-izitoast | lightweight notification plugin implemented for Vue | Notification library

1.- Should I return a promise from Vuex and then handle the data from my Vue component?

No, that's not how Vuex works.

Usually the flow would be something like this: The action gets started, which will trigger a Vuex mutation, which will update the store, causing a getter to update. Your component would have mapGetter setup for the value you're looking for and react to it.

Alternatively, you can skip using vuex altogether, in which case you could do this in the component, or a separate .js file. In this case you would return the axios call, since that is a promise.
like this: return axios.post("/api/...

2.- Can I use the library (iziToast) from Vuex? Or is this an anti pattern manner? How can I do this?

I'm not familiar with iziToast, but if there's no way to link it to a store or getter from vuex, you can likely connect them through any component, like a layout component.

for example, if you're using the vue-izitoast you could add this to any component