Is home synthesis of cellulose acetate possible, and why? Is it remotely possible to synthesize detectable quantities of cellulose acetate from materials available in common department stores (such as cotton balls, vinegar, and battery acid) and using tools available in a home kitchen (such as ovens, stoves, freezers, glass jars, and metal pots)?  If so, how would the reagents need to be pre-processed, and what would the procedure be?  If not, where do the insurmountable difficulties lie?

On a webpage titled "From Cotton to Cellulose Acetate" it is indicated that, in general, cellulose acetate is manufactured from specially prepared cellulose in a solution of 100% acetic acid with acetic anhydride.  That same site goes on to specifically state that immersing cotton balls in vinegar will only result in "cotton balls that smell like pickles".  Other sources add concentrated sulfuric acid into the mix.  And an answer to another question on the Chemistry SE indicates that household vinegar could possibly be concentrated up to 58% using fractional freezing in a kitchen freezer.

It's difficult to tell from the sources available whether making homemade cellulose acetate is completely impossible, somewhat inconvenient, surprisingly hazardous, or merely commercially unviable.
The bottom line is that this would be a very bad idea to try due to safety reasons alone, and is not realistically feasible using only household materials regardless.  

Glacial (100%) acetic acid is dangerous if not handled properly and requires an appropriate fume hood at a minimum. You also are not going to get to glacial acetic acid from vinegar, and water will kill this reaction.  

Acetic anhydride is a reagent that you just won't be able to obtain or synthesize yourself. Even if you did, it would be hydrolyzed by the water in your impure acetic acid.  

Sulfuric acid is commonly used as a catalyst, and while you could obtain this, it should absolutely not be handled by someone without appropriate equipment and training in it's safe usage.  

So again, this is just not a feasible project and there would be serious dangers along the way if you tried. I like your creative thinking process and I hope you will pursue some other, safer, home science project.