It would depend upon the full situation and state negligence laws to determine if the homeowner's policy would accept a claim. If they would then you could place a claim in through the policy of the mower. If their homeowner's insurer does not find there to be negligence on the part of the person mowing their lawn then they could deny a claim and thus you would need to place a claim through your own comprehensive coverage.
If the person mowing the lawn was taking due care in mowing the lawn and it was just an unfortunate accident than their homeowner's insurance may not cover it.
If you do not want to deal with the homeowner's company and wait for a determination if the claim will be accepted or denied then you could place a claim through your own comprehensive coverage in which case if your insurer feels the homeowner could be held responsible they could subrogate with the homeowner for the cost of repairs after paying on your claim.
It would help if you knew about your state's negligence laws and how they apply to this situation. Your insurance agent may be able to guide your or you could contact a lawyer to get this type of information. It may also make a difference if the mower admitted responsibility and said they would pay for the damages or place a claim for damages through their homeowner's or personal liability umbrella policy.
In watching television court shows regarding accidents such as this where a person was mowing a lawn and a rock flew up and hit a car, in one case the judge did find the person mowing liable for the damage that a rock caused to a passing car. In this particular instance though the person in the car won the case because they took pictures of the lawn showing that there were many rocks and that the person mowing had not taken reasonable care to avoid that patch of rocks that indeed did get spit up by the lawn mower.
So depending upon your state's negligence laws it may depend on the circumstances and the condition of the lawn that the person was mowing. If there was no way for the mower to know a rock may be spit up and it truly was an accident then they may not be found to be negligent. On the flip side if the person had knowledge that there was a rocky part of the lawn and did not take due diligence to not mow this section or to take out the rocks before mowing it then could be an issue.
If you have the homeowner's policy information for the person mowing their lawn you can call and discuss the claim with their claims department as well as call your own auto insurer to discuss making a comprehensive claim if the homeowner's claim is not accepted.