Plenty of electric lawn-mowers available. Newest models are battery powered. In general, I don't think that using a gasoline-powered lawn mower is a big deal for climate change, though.
Personally, I think that keeping my gasoline-powered lawn mower than must consume a grand total of 5L per year (I'm in the UK so the grass basically stops growing from about now until March, and although I have a large garden for local standards it's not half an acre...) makes me emit less than throwing it away and replacing it with a new battery-powered electric one. Mileage may vary, of course.
It is said that for nature the best thing to do is actually to get rid of lawns, as least some of them, and to replace them with richer ecosystems. So perhaps one option, since you seem to have space, would be to reduce the size of the lawn (which obviously makes mowing it easier) and to plant a meadow and whatever other plants.