If we want to know halacha regarding Chuppa, we need to learn what is the halachic meaning of Chuppa
See Tur Even Haezer siman 61:
What is Chuppa? The fact that he will be alone with him, even if they don't enter in marital relationship, since they were isolated in name of marriage, she is his wife for every rules.
According to this definition of Chuppa, it is not necessary to have a Chuppa opened on all 4 sides.
See Bet Yosef. In summary the Ran explains that there are two whole opinions regarding the definition of Chuppa, each of them has supportive arguments. The first, whe already soon in Tur, and which is also the opinion of the Rambam is that Chuppa is the fact that the Chatan is in Ychud with the Kalla in his house. The second opinion, with some variants is that the Chuppa is similary to our chupot, a specially reservated place, with painted linen, or a kind of succa made with roses, a place which is not necessarily discreet, and bride maids and page boys may also be present. The Rishonim and the Yerushalmi which seems to support this second opinion did not mention the need of a chuppa opened ont the all 4 sides. The point of this second opinion is that the bride enter is in his property, (in the idea, perhaps it is a bit similar to the concept of kinyan Chatser or Meshicha). I don't know nowadays how we deal with the fact that the Chuppa needs to be a property of the Chatan. A nafka mina between the twoo opinions is the legitimity of a Chupat Nidda.