A little kludgey, but it does work. This requires you have the dropbox folder open and frontmost in Finder.
tell application "Finder"
activate --brings Finder to front
select item 1 of window 1 --selects the first item in the list ; not sure how you want to select/determine the item
end tell
delay 0.5 --this is to insure you don't get unwanted typing in script editor (makes sure Finder activates before kludgey typing gets done)
tell application "System Events"
tell application process "Finder"
set _selection to value of attribute "AXFocusedUIElement" --focused element (selected element)
tell _selection to perform action "AXShowMenu" --contextual menu
end tell
keystroke "share dropbox link" --type to select contextual menu item
keystroke return --select it by hitting return
end tell
[EDIT] Here's a trick to make sure your dropbox folder is open and frontmost in the Finder:
tell application "Finder"
activate --brings Finder to front
tell me to set dbFolder to POSIX file ((do shell script "cd ~/Dropbox/;pwd") & "/")
open dbFolder
select item 1 of window 1 --selects the first item in the list ; not sure how you want to select/determine the item
end tell
tell application "System Events"
tell application process "Finder"
set _selection to value of attribute "AXFocusedUIElement" --focused element (selected element)
tell _selection to perform action "AXShowMenu" --contextual menu
end tell
keystroke "share dropbox link" --type to select contextual menu item
keystroke return --select it by hitting return
end tell
There may be another way of selecting the contextual menu, but this is what I came up with so far late at night after a glass of wine.