Does anyone know a free web hosting service that would support this Cube Cart software? I'm on a tight budget, and for now I wanted to start off with free hosting. Thanks.

Due to the advertising rules, we cannot recommend anyone here, however, can point you in the direction of www.cubecart.org that lists third party products and services.

There's plenty of hosting companies there to choose from, some of which even offer free hosting (advert spondored) if you buy a domain name. Saying that, many of the hosting packages offered are very reasonable, good value for money, so even if you have an extremely low budget, you'll probably find that they won't break the bank.