I wonder if there is any way to get free annie skins

nope

You can only get free stuff for Alistar, Tristana and Garen (Maybe. Not sure if that Twiter thing is still going on).

Technically Yi and Veigar too, since they both have a 390 RP skin.

If you ever find someone kind (or stupid) enough to gift you a skin, that'll be your best bet.

Alternatively: Don't know if the codes would even work since old codes have been deactivated or have expired, but some old, inactive accounts (like beta old) were emailed codes for the Digital Collector's Bundle (contains Goth Annie) a long time ago. If you ever get into contact with someone with one of those accounts and they are willing to give you the code and if the code still works, then, BAM, you got your free skin.

Illegitimately, and illegally you could use a modding software to get ingame skins to cover your base skin, thus giving you the skin (even with animations, sounds, and effects).

Though as I said, it's illegal to do that as you would be stealing content. Using mods to mod your game with non-riot owned skins are completely legal though, even if it's at your own risk.

I don't recommend doing this, but it would be a way you can get free skins for anyone.