You can usually run a cable through the grommet in the firewall next to the other wires. Only issue is the cable probably can't have the connector on it.
Fish a stiff small diameter wire or stick through the grommet first with the coax cable secured to the other end and then use that to pull the cable through. You'll then have to install the coax connector after the cable is run.
Alternatively you can run the cable out the door, use a smaller diameter cable (RG-58 or smaller) and route it near the door corner where it will proabably get squished the least. It tends to get in the way more but you may be able to dress it in such a way that it works for you.