Your MiFi 4510L should broadcast and charge at the same time if your charging connection is from a power source only. The MiFi will shutdown its broadcast if it detects a data connection of any kind from the charging cable. For example, car chargers and wall chargers should allow you to charge and use at the same time. Tethering your MiFi to a laptop or smartphone through its USB cable will not.
Determine if your power source has a data connection or not. If it does then you need to find a way to sever the data portion of the connection. There are steps and suggestions posted on this thread for more information:
If your power source does not have data on it then your MiFi is most likely defective and should be replaced under the warranty. Contact Verizon or bring it into a store to complete that process.