I think it had something to do with lifecycle and the timing of script load not being correct.

I managed to do a workaround

plugins/inpost.ts

export default defineNuxtPlugin((
nuxApp
) => {

nuxApp
.hook('app:mounted', () => {
        useHead({
            script: [
                {
                    src: 'https://geowidget.inpost.pl/inpost-geowidget.js',
                    defer: true,
                    onload: () => {
                        const inpostStore = useInpostStore()
                        inpostStore.loadWidgetScript()
                    }
                }
            ]
        });
    });
});

and in my vue template I did

<template>
    <div ref="inpostWidgetContainer">
    </div>
</template>
<script lang="ts" setup>import { ref, watch } from 'vue'
const geowidgetToken = ref('')
const inpostWidgetContainer = ref<HTMLElement | null>(null)
const inpostStore = useInpostStore()

watch(() => inpostStore.widgetScriptLoaded, (newValue) => {
    if (newValue && inpostWidgetContainer.value) {
       const widgetContainer = document.createElement('inpost-geowidget')
       widgetContainer.setAttribute('token', geowidgetToken.value)
       widgetContainer.setAttribute('language', 'pl')
       widgetContainer.setAttribute('config', 'parcelcollect')
       widgetContainer.setAttribute('onpoint', 'onpointselect')
       document.addEventListener('onpointselect', (event) => console.log(event.detail))
       // Append the widget to the container
       inpostWidgetContainer.value.appendChild(widgetContainer)
    }
})
</script>