Well, that response wasn't terribly helpful, as I had noted that I've been looking through the documentation and hadn't had much luck figuring it out. It seems pretty standard to respond to what is perceived as a "dumb question" by simply replying "RTFM". The thing is, it's not the Authorize.net API that I needed help with, it's how to call it with ASP. I've never tried to call a SOAP service via ASP, and I was looking for examples on how to call the Authorize.net service with ASP, and haven't been able to find that.
So, I continued to research how it's done, and I'm not 100% there yet, but for the benefit of anyone who might be stuck in my same predicament - having to work with Classic ASP - here's what I've got so far (I never found anything in the documentation that looks anything like this, BTW):
GatewayHost = "https://api.authorize.net/soap/v1/Service.asmx"
Set xmlhttp = CreateObject("Microsoft.XMLHTTP")
SOAPRequest = _
"<?xml version=""1.0"" encoding=""utf-8""?>" &_
"<soap12:Envelope xmlns:soap=""http://www.w3.org/2001/XMLSchema-instance"" xmlns:xsd=""http://www.w3.org/2001/XMLSchema"" xmlns:soap12=""http://www.w3.org/2003/05/soap-envelope"">" &_
" <soap12:Body>" &_
" <GetTransactionDetails xmlns=""https://api.authorize.net/soap/v1/"">" &_
" <merchantAuthentication>" &_
" <name>" & objRS("ProcessVendor") & "</name>" &_
" <transactionKey>" & objRS("ProcessPass") & "</transactionKey>" &_
" </merchantAuthentication>" &_
" <transId>" & objRS("Reference_Number") & "</transId>" &_
" </GetTransactionDetails>" &_
" </soap12:Body>" &_
"</soap12:Envelope>"
xmlhttp.Open "POST", GatewayHost, False
xmlhttp.setRequestHeader "Content-Type", "text/xml; charset=utf-8"
xmlhttp.setRequestHeader "SOAPAction", "https://api.authorize.net/soap/v1/GetTransactionDetails"
xmlhttp.Send SOAPRequest
response.write(xmlhttp.responseText)
I can confirm that this results in getting the SOAP response back, but I cannot (so far) convert that (using xmlhttp.responseXml or the responseText, as shown above) to an XMLDocument, which should be pretty straight-forward, but haven't had luck with that yet. Since I only really need 1 piece of data, I might just cheat and parse the response text using standard string methods.
Hopefully that will help someone out!