Annual Garden with minimal inputs?

I am starting a small annual garden in an open field which currently has perennial grasses and forbs. I am in NJ, zone 7 and the soil has a lot of clay.

My goal is to garden with as few external inputs as possible, and see what is actually necessary. I am hand digging all of my beds. I do not want to bring in any mulch or weed suppression materials, since these would count as an external inputs. I am willing to bring in organic manure based fertilizers if necessary, but other than that I do not want to add anything to the garden.

I am mostly trying to grow corn and beans. I have sowed some spring crops by hand tilling 2.5 ft wide beds. I am noticing that the top inch of my clay soil has dried out a lot making it really hard. This seems to be slowing down seedling development.

I can harvest some green manure/mulch from dead or living grass on site, and I have been covering new beds with a light layer of last years grass stalks, to hopefully retain some moisture.

I am trying to come up with alternatives or improvements on hand tilling. The tilling works to kill the grass, but the top layer of the soil dries out and hardens very easily. I have seen patches of lawn in my area filled with unammended soil that has turned compacted and lead to bare patches where nothing will grow. I am hoping to avoid this with my tillage. Also, I’d like to till as little as possible because it is hard work and disturbs the life in the soil.

The problem is that every no till method seems to rely on a large amount of external materials, such as cardboard, mulch, manure, etc. I am looking for any advice on how I can minimize tilling, or minimize the damage from till I’m in this situation. One thing I have tried is sowing corn into small piles of last year’s grass where nothing is poking through.

Welcome to Permies.

The limitations you have placed on this project have you in a box.
No till growing with lots of inputs is considered to be a big deal.
If there was a way to no till with less or no inputs, I  would expect someone to be crowing about it.

Other than using what you have at that site to create mulch, you could opt for seedlings over direct seeded plants.
Flat stones might work as mulch.
You could use your own human waste, to amend the soil, though doing that for thus years crops raises some issues.

You are gonna have to put something into growing annual crops if you expect to get something back.
Even foraging takes work , often a lot of it for a relatively low caloric yield.

You don't mention irrigation, is that available?
The right kind of tool, like a wire weeder or hula hoe might cut down on both the work load and the soil disturbance.

I think that by limiting the importation of inputs, you are going to require more time to establish the soil to grow better but it is possible.

I would go the route of green manure/cover crop on the gardening site with the intention of using the biomass to amend the soil. I'm not sure how heavy your clay is, but when I established a garden in predominantly clay I needed a bunch of organic matter to reduce compaction.

I'm attempting similar for my GAMCOD (Grow a Million Calories on Dirt) attempt - a small garden with land around that I can cut grass from. I also want to do no-dig with no external inputs, so I'm concentrating on gathering as much cut grass as I can to smother the area as thickly as I can, then I will be planting it up in the autumn when the weeds have been smothered and the soil has loosened up a bit.  

I'm also adding any kitchen waste I have under a different patch of mulch every time the caddy is full.

Here's a link to my thread on GAMCOD

And a quick video of my other half piling cut grass onto the bed.

It is perfectly possible to grow without external inputs, and it is a creditable thing to try as it avoids stealing fertility from elsewhere, or taking input from possibly more dubious sources. I think you will find that you are unlikely to be able to use the full area for growing crops however. Part of it will be better set aside for growing plants to add fertility either in the form of cover crops (living mulches to protect and feed the soil organisms) or green manures (cut to add biomass to the growing area).
One of the best references I have for this practise is Ian Tollhurst of the UK, who has been growing without animals on a small farm in the UK for a few decades and has developed a system that appears to work well from a biological point of view, although financially less rewarding.
See This site for some more information. So as well as setting aside 2/7 of the growing area for non food crops - purely to feed the soil. Ian also uses his own hedge trimmings and other woody plants to add to the soil carbon.
Helen Atthowe has a similar set up in Montana using surface mulches, ground cover and hay to continuously feed the soil organisms.

There are a lot of reasons to avoid using off site inputs.
Me, I'm cheap, so I don't want to buy fertilizer or compost.
I do grab bagged autum leaves every chance I get.
They go into the chickens composting area and onto garden beds.
Actually most of the "soil" in my beds is composted autum leaves.
I also dive dumpsters for food, some for me but mostly for the chickens, who turn it into compost.
I love getting free fertility from other places so much that I've considered starting a lawn mowing business, and I HATE mowing!

I've planted lots of comfrey and trees, and nitrogen fixing shrubbery but I think aquatic plants could be my ticket to creating my own nitrogen rich biomass with as little work as possible.

Inputs of time and materials are unavoidable if you expect to to gain a useful yield.
But we can decide how much of each and from where.

I just got a tray of seedlings from a local gardening organization.
I had joined them for a planting event, brought my own bag of runner beans and planted up that tray.
They handled the care.
I would have gladly paid money for that privledge, yet I refuse to pay for pelletized organic chicken poop.
You gotta pick your poison.