Anything you put on the lawn considered organic will decompose eventually into humus which is what you will need in that sandy soil. Go ahead and stick as much corn as you like as it is pretty weak as a fertilizer, you just have to make sure it is decomposing at a good clip before applying more. SBM is a great fertilizer but I wouldn't bother with putting that on for now as it would just spur weed development. The moss shouldn't want to hang around if it gets too dry but you can beat it up with a rake every so often and it should respond by thinning out. I would think the vinegar would have a harmful effect on the moss but the roundup probably would not (just my guess).

Whatever you use to kill the moss, organic or non-organic, will not prevent it from coming back. To do this you need to change the conditions in which it thrives. The key is to give your grass the competitive advantage vs. the moss.

1. Moss likes acidic soil

Have your soil PH tested by your local extension office. If your soil's PH is below grass' preferred PH range (5.8-6.5) then you need to amend per their recommendations. This will not kill the moss but is one part to swinging the advantage towards growing grass.

2. Moss likes shade

There is not much you can do about this except see if you can prune the tree to add sunlight. When planting grass be sure to use a grass that does well in the shade.

3. Moss loves moisture

This is the hardest part to correct but also the most rewarding. Compacted clay can actually hold a lot of water and is one of the most difficult soils on which to grow grass. Moss has very shallow roots, which is why it grows well. Your ultimate goal is to increase the depth of humus and have grass who's roots reach deep into the soil. This way, water will run deeper into the soil (where the grass' roots still reach) and still allow the top 2 inches to be dryer, prevent moss from thriving.

Clay is also very hard when dry so you may want to put a soaker hose on it to make it easier to work with when re-mediating.

Since you are starting from scratch I would till in a few inches of cured compost. Sand will make it worse, think concrete. If you don't want to do that you should core aerate to allow access lower in the soil. Rent the machine or buy the hand aerator that actually removed the soil rather than using the golf shoe approach, which will actually compact it more. Don't be afraid to create a lot of holes.

After core aerating you need to add organic matter (compost) and microbes to change the soil structure. Using a surfactant (johnson's baby shampoo) in a hose sprayer to change water tension and open air pockets in the clay will help speed things up. Follow with humic acid to add microbes. Use Kelp when seeding to improve root growth. Cut grass high and leave the clippings on to add organic matter once the lawn is established. Top dress with compost once a year for a few years to add organic matter. Water deeply and infrequently. I would be careful with roundup or any herbacide before seeding. Raking the moss initially should be enough.

"Moss likes acidic soil"

I keep seeing people say this, but it does just fine in the part of my lawn that is shaded by the fence and overwatered by my neighbor. My soil's pH is about 7.5.

I think the key factors are the shade and the moisture.

Moss likes extremes of PH, it is usually acidic but it can be soil that is alkaline. It just shows that one should know their soil's PH and adjust to the range in which grass likes most.

Moss likes shade. There is a relationship between moss and pH but it isn't causal.

Areas shaded by stands of several older trees tend to have more fungaly dominate soils which in turn tend to be more acidic. However the fact that those areas are often mossy has little to do with the acidity levels. Acid soils with lots of sun exposure will never grow moss. On the other hand you can lime shade till the cows come home but moss can still grow as long as the trees thrive.