I'm not that familiar with LabVIEW web services but have you checked the security access permissions on your web service VIs (and the application web server) to allow access from other hosts?
In order to access resources across domains from the browser you probably need to enable Cross-Origin Resource Sharing (CORS) in the webservice. The process for enabling CORS is described in the Using CORS: Adding CORS support to the server. I have not tried this myself in the LabVIEW Webservices but you can probably use the Set HTTP Header VI in your resouce VI to set the HTTP Header field "Access-Control-Allow-Origin" to the value "*" to allow any domain or specify the allowed domain.
See the Access-Control-Allow-Origin description in the tutorial and the Configuring HTTP Headers LabVIEW Help Topic. One thing to note from the tutorial is that additional headers are needed for requests other than HEAD, GET, and POST.
