In fact many, though not all, US Wal-Marts permit overnight parking - even for RV campers. I suppose in theory those which permit it might be more likely to not notice or care if your car were there for a week, but I couldn't say for sure. You can google Wal-Mart overnight parking and find quite a bit of discussion about it - probably there have been others who have actually done what you propose and you could see what their experiences were. This site is one example - it lists those Wal-Marts that
don't allow overnighting:
No Overnight Parking at Walmart ? Walmart Locator
Still, even if you confirmed that the store would allow you to do it, I'd also wonder how much you're saving to make it worth taking the extra step and hassle of doing this.