It could be the alarm system, the ignition switch, battery, battery cables, terminals,or the Power Control Module.
We can do a couple of things.
Test the battery for 12 volts and do a load test, the battery could be to weak when under a load and will not crank the engine over. You can have the load test done for free at autozone. This is the first step.
It battery test is great, check the cables and the terminals.
Pull the fuse for the alarm system and see if it is blown or not connected all the way in.
Check your manual for the location.
Check the relay box next to the battery, check to make sure all the relays are pushed down, you can check the starting relay, or the ignition relay and switch it with the horn relay if they are the same relay.then try the vehicle.
Ok after all that is checked out, do you know any one with a diagnostic scan tool? This tool will tell us if the modules are working and can reset them or tell if they are bad.
If you do not know anyone, and you want to do the electrical test yourself to the ignitions switch and the Power control module let me know and I will send you pin outs and the locations to test for power. Lets hope we find the problem before that.
Please let me know how the testing goes and on to the next steps.