Even if Web edition did support Log Shipping (see icelava's answer), you can log ship without a mapped drive. We do it all the time. You just ship to a UNC share (\\server\share).
You can set it all up using the Log Shipping wizard in the SQL Server Management Console, no need to script it if you don't want (or, it will generate the script for you I believe and you can just modify it appropriately for each instance you want to run it on).
Looking at more detail in your question, sounds like what you mean is "How do I log-ship without using a share" (i.e. your 2nd web server is not on the same network). AFAIK, you can't, but what you can do fairly easilly is set up a VPN between the two. If you own both the servers (or they are VPS's), then setting up a VPN is as easy as installing a program called Hamachi at both ends and then you can communicate between the two using the IP address as your sharename (e.g. \\5.x.x.x\share).
Edit: I see that your question now reflects the UNC path part of the answer