Lis, you kind of already know some of my thoughts on shoes. Even if you don't want to spend a fortune, I'd still be taking her into a shop to at least get her feet measured - you don't have to buy straight away, and if you're looking for an excuse, tell them that your mum wants to buy the first shoes, so you're getting her measured & choosing a style so that your mum can come in to purchase later that day (or whatever).

Also, I think flat feet a normal for babies/toddlers, but again, if you want to check, ask the person who measures her up. The Clarks fitting people I've dealt with have been very helpful.