I used desktops with internet access in the STL Admirals Club and MIA Admirals Club.
Not sure about LAX and ORD Flagship Lounges. I think you need to visit the Admirals Clubs for internet access.
LAX flagship does not have a business centre, so you have to use the AC (which is right across from the F/S lounge.
ORD flagship has a work centre with 4 (IIRC) computers in it.
If you are making multiple trips in one 24hr period and have a laptop, it might be worth logging on to T-mobile and buying a 24 hour pass ($9.99). It works in all AC and f/s lounges and can be easier than waiting for a computer. However, I find that I don't often have to wait too long for a computer.