How can antibodies in rapid antibody tests be kept at room temperature without denaturing?
Antibodies have no problem with room temperature. Naturally, antibodies expect to survive for many days or weeks at body temperature, which is much higher than room temperature (at least, mine is).  

-20°C is recommended for very long-term storage (many months or years) of antibodies, but they are typically very stable proteins and don't denature very readily at all. Storing them at room temperature can be done for a long time -- certainly weeks, in my experience, with the only real concern being the possibility of bacterial or fungal growth if the antibodies aren't sterile.