I don't think the heat would affect the fabric at all for that short period of time. What would worry me more is the family seeing all those boxes!
You can scrounge cardboard boxes from stores, but it's a whole lot easier to simply buy boxes as then you can get all the same size so they stack easily. Walmart has some sizes; storage places and moving stores (such as U-Haul) have a lot more sizes but charge quite a bit per box (as I recall, $4-5). If you have time, join a Freecycle list (Google Freecycle to find one for your area) and post a request there for moving boxes.
Alternatively, you can buy plastic boxes at Walmart. I like to store my fabric in clear plastic boxes that stack easily, but you may want to look at opaque boxes so the family cannot see the fabric.
Be careful not to get huge boxes, as fabric becomes very heavy. Even for permanent storage, I prefer smaller boxes because I can move them around much more easily.
Personally, I would find smaller boxes, find a spot in the garage, and move all the fabric by myself when no one is home.