If your web service is more complicated and you're more comfortable writing the code to access it in .NET, then you can use COM interop to call into your .NET assembly from the classic ASP page. Here's the type of code you might then use in your ASP page:
// Create an instance of the .NET assembly
Set service = Server.CreateObject("MyDotNetAssembly.CustomerService")
// Invoke the method on your object that calls into the web service
service.GetCustomer customerId
// You can expose the results of the service call as properties on your object
Response.Write "First name: " & service.FirstName
Response.Write "Last name: " & service.LastName
// Clean up your object reference
Set service = Nothing
For this to work you'll need to strong name your .NET assembly, register it in the GAC using gacutil, and finally register it for COM interop using regasm.