Does anyone know if it is legal to install your own Fire Suppression Sprinkler system?

Some states require license for this type of work. Not an exempt activity such as homeowner working on his own home. Contact local officials.

If it was your own, private home, you could. But, is sounds like you're talking about a commercial building (apartments?). In any case, you'd be dollars ahead by having a licensed professional do the job. You'll need a backflow prevention device to keep the water isolated from the municipal water system (even though the lines are dry, after activation, they will be wet). I haven't read the code, but is there an interface with the fire department to notify them if there is water flow in the sprinkler system? And there's also an interface to set off the alarm in the building if there is water flow. The system has to be set up in zones so that FD will know where the fire is.

It's a lot more than just hooking up pipes and sprinkler heads.

No. You need to be licensed. Your insurance would not cover damage if a fire burnt down the building unless the sprinklers were installed by a licensed installer. The Fire Marshall will not even inspect the lines.