bjmoose514, it appears that it is either a new irrigation system or else it is a new problem. If the sprinkler company could find nothing wrong, assuming you demonstrated the problem to them, I would conclude that you hired the wrong company.
It would appear that the sprinkler and the house share the same supply and the flow of water (gallons per minute) is not enough to supply both optimally at the same time. You could:-
Operate the sprinkler system in a different time frame (early am?) when there is minimum demand by household.
Seek to increase the supply at the source (meter). This can involve Municipal employees and hydraulic considerations like size of pipes, psi, gpm etc. Then there is the over riding consideration $&C.
Operate the system off another meter. Same considerations as second choice.

Thanks for input. I have the city coming out tomorrow to check the water supply. I will keep you posted.