Nothing wrong with Red Oak for a project like this although Elm, Maple or Hickory would be better.  This is essentially a board bow with strange ends.    I get small, thin pieces of all sorts of wood from Ebay all the time; you just have to keep looking.  You certainly don't need exotic woods, I just happen to have them around from building dulcimers.  This Elm came from Half-Eye.  Thanks Rich!!!  
One of these days I'm tempted to laminate several thin pieces of different woods together to make up the main limb - say 5 pieces of 1/16" thick Elm, Maple, and Walnut.  Then when I tiller away at the belly, the different layers would revealed...
The wrappings, drenched in Superglue, add some reinforcement at critical areas.  Think of them as position specific "backings".