You know it's funny . . . I have totally accepted a job via e-mail, but I have never hired someone only by e-mail. Interesting.
I think indeed you should e-mail the first party and give them a deadline of 24 hours, and then move on. But in the end, if they are not responding within 24 hours, I think you can move on to the next person. Heck, even if I called someone to offer a job and left a message, if I didn't get an answer in 24 hours, I would on. Especially in ASAP situation.