Orders placed, if stored in a file on the server will require a little space, but it is minimal. Credit card info should never be stored on a server, it should be sent, immediately and encrypted to a secure location (the bank, merchant account provider or your encrypted email account OFF of the host machine.)The space required really depends on the site itself.
Don't try to pinch pennies to tightly, if the site is 30 megs, get a minimum of 50 megs.
Most hosts offering "free" shopping cart software are only offering the shopping cart for free, then you must pay for the merchant account. Sometimes, this is not the best deal available for that particular cart/merchant account. You just have to shop around.
The best scenario, IMHO, is to get your space from a host that will work with you and allow you to install whatever scripts you choose.
There are many good free shopping carts, but you still need a merchant account, not free.
Most of the free carts are free because they are designed to work with specific merchant accounts. (It's not so much the cart as it is the bank or other vendor being able to conect with the cart you have chosen.)
BEefore you proceed, find out if your client has a merchant account. Talk to the representative and be sure you proceed with a compatible cart.