How are air-core inductors typically constructed?
It is as simple as using a form or armature over which you roll your wire.  Once it's on the armature and wound up you can glue it together or even paint it with an epoxide paint that will give it strength.  However, you will have to account for heat production.  In some cases you might want to leave the coli on the the winding armature, if you've chosen the \$\mu_r\$ to be ~ =1 this could also be save.

If you are using litz wire you'll probably have to leave it on the armature.

There are companies that sell brackets and clips for this sort of thing.  the term you will be looking for and here is a link to a page giving hardware.  there are lots of competitors.