For me, Opera works with Weby from Launchy.
Maybe it's a matter of versions, but typically, if I try with "lets search something", it opens "http://www.google.com/search?source=launchy&q=lets%20search%20something" in Opera, in a new tab.
I'm using Opera 10.0 beta 3, and Launchy 2.1.2
Note that if you use a 10.0 beta like me, opening a link (execute) with Opera being initially closed, will probably not work if you use "save previous session". For me, it opens Opera with the restored session, but not with the new link. But this is a known issue from the beta. It works if you open a link, with Opera already opened.
This opening link from a closed Opera worked with the 9.x versions, so I guess it will work again, it's only a temporary bug during beta. In general, this problem happens too if you try to click a link in some other software, when it will require Opera to open, it won't work on first try.
For now, what you can do is to write in Launchy your Weby request ; it will open Opera, without your wanted link ; call launchy again and press enter directly, to send again the same request ; this time it will work.
Keep in mind that this is a temporary issue from Opera beta, and will most likely be fixed in the final version.
If the problem is in Opera 9.64 too, then maybe I'm wrong, and I remember badly. Maybe I was not saving sessions then, I'm not sure. Anyway, I'll submit it to Opera, restoring session or not, it should open the link.
Update on the problem
The issue with Opera, which was not opening links from external, on launch, has been solved in the Release Candidate version. You can find it here : http://www.opera.com/browser/next/ (or use the direct link to installer).
I checked with Launchy and its Weby plugin, it works now correctly, even if Opera is closed.