You'll need a scriptable browser such as phantomJS. You could also use an IE com object if in windows.
Problem is, as soon as you automate the process, splunk will make website changes and break your script 😉
I have seen some people around here who have done it though. It is possible.
This seems to get the latest version of 6.4.0 without having to add the build number;
Had the EXACT same issue where I wanted to automate Splunk installs but didn't want to use expect scripts for variable inputs or have to hard code versions and keep the scripts updated - YUK!
You'll want to customize this but to just download the latest version, see URL's below. NO IDEA how long they'll last or if they'll break though. I have case open with Splunk on this now and just found this by tinkering with the URI's - fairly straightforward.
Invoke-WebRequest "https://www.splunk.com/bin/splunk/DownloadActivityServlet?architecture=**x86_64**&platform=**windows..." -OutFile splunk-latest-installer.msi
wget -O splunk-latest-installer.rpm "https://www.splunk.com/bin/splunk/DownloadActivityServlet?architecture=**x86_64**&platform=**linux**..."