Depending how you have Snagit set up, either from the capture window or from the OneClick "manage presets" you can edit one of the presets or create a new one and under the "Share" field select FTP and than click on the settings cog to input your ftp info
Add your FTP settings
I don't use this feature myself so I do not know of the ability of coping the url.
I save my images/videos to project folders on my system and then upload them using Filezilla FTP. Sorry :(