I hired a contractor to do some work, the work is complete and I got a bill.
The amount is $800. On the invoice at the bottom they wrote in ink make
check payable to "john smith", the owner. Not the name of the company.

I guess they are doing this to avoid taxes.

The issue is, I would like to make the check out to the business's name as
this work involved a permit and was done over a dispute with a neighbor so I
want a record incase there are any further legal issues between me and my
neighbor. Should I just make the check out to the companies business or to
the owner?

Hmm. A couple of ways to go on this.

You could call him and say that if he wants this to be off the books,
then offer me a substantial discount.

or, (and I'd tend to do this for the reasons you cited)

You could call him and just let him know that you don't feel
comfortable doing this off the books because of permitting and other
legal reasons, and let him know you'll be sending the check in the
company name.