Is this an aftermarket car alarm? If yes, the following applies:
Please post photo of alarm control box if you can.
Aftermarket car alarms typically interrupt power to the starter (or starter relay) by intercepting the start lead from the ignition switch to the starter solenoid or starter relay. Find the wiring loom from the ignition switch. Follow this loom from the switch till you find the location where the start wire has been cut & wires are spliced in. Remove the alarm system wires that are spliced in the start wire. Reconnect the start wires. That should bypass the starter interrupt circuit from the alarm.
If this is not an aftermarket alarm, then the cure is different.
You can try this:
The video explains the concept, and give a location to intercept the 'disable' wire (purple w/yellow trace - 18 gauge). As you can see from the video, this is a difficult location to work. I prefer intercepting the same wire in the wiring harness from the door to the door frame inside the car behind the drivers kick panel.
I have had to do this with the stock jeep car alarm module on many jeeps (Cherokees, Grand Cherokees, Comanches). If this fails (& sometimes it does) your other option is carry a spare key, and use it to unlock the door from the outside while starting. You will have to leave the key in the door while driving to keep the alarm defeated. I have used the key in the door method to get jeeps back to the shop so I could fix them. Good luck. I hope my English translates well enough to help. (BTW this is not my video - just one I found on YouTube)