No. Static groups are groups of publications that have already been imported, and a set of static groups is saved as a property list in a BibTeX comment when you save a .bib file, as you can see if you open the .bib file in a text editor. Search groups and web groups only exist when BibDesk is running; they are not saved in the .bib file like static groups.
However, using AppleScript there are probably various ways to automate the opening of search groups and web groups. For example, if you wanted to open several search groups and web groups with one command, you could create an AppleScript that uses GUI scripting, something like the script below, and then save it in your BibDesk Scripts menu, and then assign it a keyboard shortcut in the Keyboard System Preferences, so you can open all the groups with one keyboard shortcut.
if application "BibDesk" is running then
tell application "BibDesk" to activate
try
tell application "System Events" to tell process "BibDesk" to click menu item "Google Scholar" of its menu of menu bar item "Bookmarks" of menu bar 1
end try
try
tell application "System Events" to tell process "BibDesk" to click menu item "Library of Congress" of its menu of menu bar item "Searches" of menu bar 1
end try
end if