Crutches seem to be one of the more rare components. They can be located in places such as Our Lady of Hope Hospital, Rivet City (Two are located inside Pinkerton's lab by the surgery table) and the yao guai tunnels. They can also be found in many raider camps. In Minefield there are a couple in the Gibson house, one in a mailbox just up the road from from the Gibson house and a couple more where Arkansas snipes you from. There are certain locations in The Pitt and Mothership Zeta where these can be found. You can also find one inside your Tenpenny Tower suite if you purchase the Science theme. You also get access to several other crafting components by choosing this theme as well. There are also 1 or 2 in a cabinet in Tenpenny Tower in a suite on one of the lower floors. Two can also be found in a tunnel, by the undetonated nuke in front of Vault-Tec headquarters, (the right side of the tunnel). Several crutches can also be found inside the tents of the Point Lookout location disaster relief outpost.
The crutch itself is missing a handle, which would make it very difficult to use for the purpose it was originally made for.