I am a student starting law school in the fall. I've been denied
credit-based loans for "insufficient credit history" and I've been
denied using my parents as cosigners due to their bad credit.

I have already applied for and been granted the maximum federal
Stafford loans. I'm looking for private loans to bridge the gap
between the funds I have (my scholarship and Staffords) and the actual
cost of attendance (rent, etc). I would prefer to avoid going outside
my family for a cosigner if possible, so I need other options.

Is it possible to get a private law school loan after being denied due
to an insufficient credit history?

Doesn't your school have a fin aid office? Most schools are allowed to
loan you up to the cost of attendance (which includes rent, etc.), and
they don't even do a credit check on you. Maybe that will help.

Shop around for different loans. Each program has it's own set of
requirements.  Also, check out the TERI Loan program--I think they are
generally more forgiving for students with limited (not bad) credit
history.