Please note that Transmit Disk is on longer supported as of Transmit 5.6.0. Details
Here’s a script that will mount a particular favorite. (Transmit must launch in order for a disk to be mounted via AppleScript):
tell application "Transmit"
set myFave to item 1 of (favorites whose name is "My Favorite")
tell current tab of (make new document at end)
connect to myFave with mount
Just save this script in AppleScript Editor (which comes with Mac OS X), then open the System Preferences app, click on Users & Groups, click Login Items, then add the script as a login item.
You can view Transmit’s full AppleScript dictionary in AppleScript Editor by clicking File menu > Open Dictionary, and then choosing Transmit from the list.