Collector said:
How do you get rid of dandylions without any chems?
Pull them up in moderately-damp-soil weather, getting as much root as possible.
or
Use any of a variety of weeding implements to aid you in doing same.
or
Use fishtail weeder or bread knife (or etc) to cut the root as deep in the soil as you can reach, and remove.
and
be real nice to your turfgrass so it grows as densely and lushly as possible, and try to avoid doing things to open sparse/bare patches, to make it harder for new seedlings to establish and to help the grass outcompete what seedlings DO get in.
Also -- and this is my major tactic -- remind yourself that
dandelions stay green pretty well during droughty weather, therefore the more dandelions in your lawn, the better it looks (at least from the road) during a hot dry August

(Also true of many other lawn "weeds".)
Oh, and sheep really love dandelions. I am not convinced they make a permanent difference but certainly the dandelion flowers/leaves vanish within the first 24 hrs or so when I put the sheep on the lawn to graze. Admittedly not always an option.
Finally, my young yellow lab also loves dandelions -- he runs off with the ones I weed up by the root and eats them, and has started actually harvesting his own. Unfortunately, lacking opposable thumbs he tends to just chew the crown off so it is by no means permanent removal, but hey, it's SOMEthing. No, you cannot borrow him
Pat