Hi Rachelle,
You might be able to get a 'facial solarium' instead of a full-size tanning bed. Phillips make one that sells for about 50 new in the UK. It would also be a lot easier to store! I have one that I've used for photo/printmaking but not yet for cyanotype, however I believe it works fine. (I'm going to set mine up with my enlarger timer to make exposures more repeatable.)
Hope you get lucky and find one cheap!