Most important call you need to, except if it's pure AJAX, is event.preventDefault() to stop the default form behaviour.
One simple way of doing it is
isValid(event) {
event.preventDefault()
let valid = true
['firstname', 'lastname', 'email'].forEach(name => {
if (this.$refs[name].value === '')
valid = false
})
return valid
}
You could expand it to look at type to validate email etc and give user feedback.
I realize this is more basic JavaScript than actual Vue. So feel free to provide an example so I know what kind of context it is.