Oak Brook is very nice small residential population and a huge and fairly upscale shopping mall plus McDs HQ and some other major offices and has some pricey suburbs immediately close by. Clarendon Hills is nice and has commuter rail links to the city -- nice for trips there with the DCs. Darien, south of Clarendon Hills, is also nice, and a bit to the south and east are LaGrange and LaGrange Park. Hinsdale is vair naice, vair pricey...
You could live in Oak Park, about half an hour/forty mins rush hour commute directly to the east, which might be smart to consider as there are more rentals available there, and it really is a very liveable community, or River Forest, and your DH would have a sort of a reverse commute. There is also Westchester, again east of Oak Brook. (The names of the streets in Westchester are 'English' -- Buckingham, Newcastle, Portsmouth, Hull, Manchester, Bristol, Kensington, Windsor
). It's a postwar suburb whereas Oak Park, River Forest and Riverside, which is also a possibility perhaps, are older. Not too many apartment rentals there, whereas they abound in Oak Park. Another place you might consider is Elmhurst, to the north of Oak Brook on Route 83. It's a nice family suburb, good schools and amenities, nice downtown area with a small cinema, and commuter rail links to Chicago.
Don't go near Villa Park, Addison, Bensenville, Franklin Park, Northlake, Maywood, Bellwood or Hillside, Melrose Park, Berkeley, Broadview too, and while Forest Park is nice for evenings out, good bars, good restaurants, etc. I wouldn't live there even though it can be affordable, mainly because of the schools. The pool there is very nice though..
I'm not really familiar with points west of Oak Brook Downer's Grove, Lombard, Woodridge, Glen Ellyn, Wheaton I have a suspicion renting would be difficult there and I also think you would both need cars out there. I could be wrong..
Staying fairly closeby is advisable as traffic is always heavy and it would be difficult for everyone if your DH had to spend too much time on the road. A suburb or location where your DCs can walk to school might be really handy for you.