No, we do not use inversion. 1. That part is not even the question — the question is simply “do you have X?” The clause “that you would like to share” merely specifies what we mean by “anything.” The phrase "anything that you would like to share" is the object of "Do you have?" 2. Even if that clause really were the question, we do not use inversion in embedded questions (questions in subordinate clause). For example: I would like to know where you live.

I think in this case "you would" is correct. In fact you already made the "inversion" of the question saying "Do you ..".