I don't have central air and my house is 82 degrees during a hot day. My crawl space is at 72 degrees so I figured I could turn on the blower to cycle some of that cool air in my house. I turned the switch to Fan On and nothing happened. I looked into my thermostat and it does not have a fan relay. Would changing my thermostat give me the option to cycle the fan or would I just be wasting money?

What kind of furnace do you have. What kind of thermostat... Some certainly can (mine can for instance), I'll guess there are some that might not.