@Nafario My code looks different because I'm using debounce as shown in @martinszemitis post above. But basically wrap it in a if /else statement. I have it mostly working with the following.

watch(search, debounce(() => {
    if(search.value) {
        Inertia.get('/users', {search: search.value}, {
            preserveState: true,
            replace: true
        });
    } else {
        Inertia.get('/users');
    }
},500));

If you type in the search box and then backspace till it's empty, it will make the ajax request to just 'users' with no query string, getting rid of the empty '?search='

This seems good enough to me but not perfect because if a user gets there by a link like 'mysite.com/users?search=' then it doesn't remove the query because we aren't firing the ajax calls yet.

Update: I used onMounted hook to remove the query param if the user visits with a link containing the empty query.

onMounted(() => {
    if(search.value === null)
        Inertia.get('/users');
});

Be aware if you check the wrong prop or variable you can easily get into a infinite loop of ajax requests.

Filtering, State, and Query Strings