My dd is same age as yours OP and she still doesn't own a pair of shoes lol!
I took her to work last week and everyone was asking me why she wasn't wearing shoes. To be honest she's never really needed them yet. And she has never kept a pair of socks or a hat on and I don't think I'll going to have much luck with shoes either. She likes to be natural.
If I was in a place where they were needed such as bindis then yes I would put them on but I haven't been in a situation yet where they have been required and running around barefoot outside certainly has never been an issue yet.