Cheapest? Food waste from a grocery store (produce) or restaurant. Will look awful, smell awful, but you'll get fantastic results. ;) Oh you meant something more neighbor-friendly? :)

Where in the world are you?
Are there deciduous trees around that you can gather the leaves they drop each year? Very good source of organic matter and mostly free although landscapers are paid to haul those leaves away so they can compost them and sell that compost back to the homeowner the next spring.
If your states public health laws allow collecting waste from grocers and resturants that might be a source, but check about those laws first.
Tree cutters may dump the wood chips they create when removing trees if they need to haul them away and pay a dumping fee to put them in a landfill, some of which then compost those wood chips and then sell that compost to the landscape companies the sell it to you.
Then there is the vegetative waste you produce in your yard and garden and home, that many people pay someone to haul away to landfills where that material creates pollution.
Manure, when mixed 1 part manure to 3 parts vegetative waste, can be a source of organic matter but not a good one by itself. Manure however can be rather expensive and may not be free.

I agree with kimmsr. I throw any kitchen scrap in my yard. And I've been known to gladly take leaves from people's yards. As mentioned before, someone is either going to take them up and burn them or have someone haul em off so get em. And don't forget that mulch mowing is also adding OM to your lawns soil.

My neighbor thinks I'm crazy for taking his leaves in the fall. I mow as much as I can into the lawn as possible in the fall but I've found it works better for me to bag them up and spread them out in the spring when the grass is growing better. I just ration the bags out with each mowing throughout the spring.

You don't need it but if you get (free) manure, be sure it smells wonderful before you put it on living stuff.

Best source of om in the soil is roots. And then if they are fertilized with organic fertilizer, that helps, but roots are the best. So just grow grass and fertilize it with organics rather than synthetics. Keep away from insecticides because they definitely harm the microarthropods in the soil. If you want to apply organic monthly, or weekly, that might help improve the OM faster. You'd have to be doing constant testing to really know if it helps, but you would definitely see the improvement in your grass. And the healthier roots in the ground will help improve the OM.

An organic "fertilizer" that is fairly common throughout the USA is Milorganite which say it will add organic matter to the soil, which it will. However, the amount you can apply is so insignificant in the quantities you would apply that you will never add enough OM to be usefull and if you put down enough of that to make much difference you would burn your grass. Even if you put similar products down weekly (probably cost prohibitive) you would not put enough on your soil to make much difference.
Animal manures are pretty much the same. to get enough on your soil to make much difference in soil organic matter you would need to apply so much that it would burn your present turf. Make compost from that animal manure and vegetative waste and apply that 1/4 inch at a time is much better then applying just animal manure.

You might head over to the soils forum and look up "lasagne Bed" or "Lasagne composting". It lets you use time, effort, & ingenuity instead of $. Works good too.