@boghy933 I would create a separate vue component for the form and then you can use this component on the various websites that need the form.
if some of the functionality changes between sites, make these changes reliant on configuration and you can put this configuration in a .env that will be read in by your vue app when it is compiled into your assets, or you can include it as js variables and read them at run time.
you should also set up CORS for your laravel api so it can only be hit by requests from your vue form, and that will prevent any unsolicited traffic overloading your api.