In general, all of the big chains will have SOME London hotels where the basic room level can fit 4 people.
You will be limited to a handful of hotels from each chain, of course.
In terms of numbers, IHG is the biggest chain in London followed by Hilton and Carlson (Radisson Blu). I have redeemed with all of these chains for rooms for my brothers family of 4 when he was visiting London.
Club Carlson is particularly good as they offer family rooms that sleep 4 for additional points at many places. Park Plaza County Hall and Park Plaza Westminster Bridge are well placed for many top tourist spots.
The easiest thing to do is look on the websites for each chain for a random night a few months ahead, select reward nights, select 2 adults and 2 children, and see how many properties appear.