We are a military family and don't have family around either. I was 7 1/2 months pregnant with my second daughter when we moved here, so I knew NOBODY....so I had to do my homework quick in order to find care for my then 20 month old. I researched Home Child Care Providers in the area and did interviews and then had a "mock" 2 hr. stay with one provider to see how my daughter would do. She did great, but still felt a little strange...if it would have been a middle of the night labor. My husband and I ended up paying to fly my mother-in-law (who doesn't work) out here...just in case. Mind you, my mother-in-law is practically a stranger to my daughter (since we are a military family, we are very rarely around family)...but felt somewhat more comfortable with the idea of having her be here than a total stranger. Ended up my water broke in the middle of the night. My husband took me to the hospital (while my mil stayed at home with my daughter), her dropped me off (I was 5cm) so he came back home, got my daughter up from bed, feed her breakfast, etc., then headed back to the hospital...to which he made it just in time because I delivered 25 min. later. :) After I delivered, he stayed for about 30min. and came back home. So all in all, he was with me for about an hour. A lot of people think we are crazy for that, but our first daughter was priority....NOT him being with me and baby. Baby and I were fine...I'm a Labor and Delivery nurse, so I know all the things to expect etc., so I was totally fine, if by chance he did miss the birth.
Anyway, all that to say.... honestly, if I were you....I would NOT feel guilty about calling your friends in the middle of the night (should you go into labor). Your friends offered, because they want to help you. They KNOW that labor can come at any time of day, so if they didn't want to take your child at 2a.m. then they wouldn't have offered. Also, don't worry about your daughter being tramatized. Kids are VERY VERY resilent...it probably won't phase her one bit...especially if you choose someone that she knows VERY well and feels comfortable with. Or another alternative is having the friend come to your house. I think the sitter "on call" is a great idea...but make sure your daughter knows who that sitter is...just in case you have a long labor and your daughter wakes up with the sitter in your house.
Good luck to you. I'm sure you will figure out something. Try not to worry about it...it will all fall into place, just have your back up plans ready!