We live in South Austin (zip 78735). I am looking for a service that
will pickup and drop off my 3 year old to and from daycare everyday
which is less than 7 miles one way. The nanny services don't seem to
do this kind of activity.

One additional constraint is that I like the have the same driver
every day - that way I can verify that I entrust my kid to a reliable
driver.

Is there a local organization that does this?

Maybe you need something like Rent a Grandma.
Have you asked the local newspaper?  
Maybe your or another church knows someone.