I've rented in many different places, and always mowed the lawn as a default without even considering that it might not be our responsibility. I view it as typical upkeep, like vacuuming. I had one rental where the LL showed up a week after we moved in and mowed, and I was stunned. I assumed it would be our responsibility. So, I guess it depends on the LL.
Since you don't have a mower, and it's not in the lease, it seems reasonable that he at least provide the mower.