Why call a foundation repair company vs a general contractor?
The house can’t lean away from the chimney so the chimney would have to lean away from the house. The only thing that can cause a chimney to lean away is failure of the foundation.
Telltale signs that you would look for.
Gap between the wall of the house and the edge of the chimney. As you look at the bottom there would be a small separation then as you look towards the top the separation gradually gets larger and larger. The chimney is now out of plumb rather than being vertical along with the vertical of the house.
A short term concern is water leaks. With a separation between the sheathing and chimney, you have opened up the opportunity for water to infiltrate. If there is a blowing rain storm, you are going to get water blown in behind the chimney.
The top of the chimney is flashed against the roof to prevent water from going down under the shingles and leaking into the house. When the chimney pulls away enough to pull the flashing, then you are going to get water directly from the top of the chimney. Water damage happens more quickly than a foundation failure causes severe structural issues so the first thing you will notice will be water damage.