S.Y.
How wonderful that you are doing this. My sons have quilts that their grandmother made (25 years ago) and they treasure them.
I don't know of anyone personally in the Austin area who does quilts, but I can tell you that the Lutheran church is notorious for having quilting clubs. I'm sure other churches have the same thing, and they could put you on the right path to finding someone who does quality work.
I would just call and ask the main office if they have a quilting circle and could you please have the name of one of the members.
You might also try googling "texas quilts" or "quilts texas" or "quilts austin texas" (you get the idea) and see what comes up. I think your biggest hurdle will be finding someone who does quality work. If you can find someone who will hand-make a quilt for you, you might want to ask them if you could bring them some of the children's old clothes or maybe some pieces of a favorite blanket to incorporate into the new quilt(s).
Good luck!