Does there exist a chemical-free, homemade substitute to deli?
Roast beef is relatively easy to make at home and is much more delicious that store-bought.  Here is a good roast beef recipe from the New York Times.

There is a Japanese method of cooking chicken called "Torihamu" where the goal is to make the chicken the texture and flavor of deli ham.  Here is a little more information and a recipe.

A lot of the flavors of deli meat are dependent on salt.  A deli meat without salt just does not taste the same.  Remember that when making it at home, you will always be using less salt than a manufactured meat.  Celery salt may also give you the flavor you need.

There's also a lot of good recipes like this DIY Homemade Lunch Meat.

Here is a video for making deli ham.  The trick to that deli ham texture is cooking gently in water and then pressing.

I hope this was helpful!  Best of luck to you.