The temporary crown on the back most lower tooth has broken. can I go without it until the appointment for the permanent on 9/21

I'm just seeing your question today, but it looks like your appointment was today.  I'll still answer this question in case someone else comes along with a similar question in the future.  Breaking your temp is not a problem, as long as you're gentle with that tooth, stay away from chewy or hard foods and are not experiencing any pain.  As long as you are able to protect that tooth, you are fine.  Good luck with everything and I hope you love your new crown.  You can click "save" to follow me on RealSelf if you have any other questions you'd like for me to answer.

I would highly discourage waiting this long to get a temporary crown replace.  Temporary crown serves many purposes, such as keeping your crown prep intact and managing the soft tissue around the prep.  Although it is still possible for the dentist to cement your permanent crown on 9/21, it might be more difficult than if the temporary crown was there.

Temps are very important.  They are placeholders for the future final crown.  If you are missing any part of it, it can cause teeth around it to move and the final crown wont fit correctly.  If I were you, I would try recementing the crown back together

yes you need to go in to have the temporary crown repaired because if you don't contacts can shift and opposing teeth can erupt. Also the gums can tighten around the tooth to make it extremely difficult to fit the crown being fabricated.