I checked back thinking this one would have been answered quickly but the thread is going in a number of directions, some wrong.
Sears never took on manufacturing bicycles themselves but as a major retailer, they had enough clout to have things their way.
At the height of the prewar balloon period, they purportedly had an in-house bicycle group that worked with the manufacturers that supplied them with bikes to design or customize models to make them more exclusively “Elgin”. The Bluebird, and later the Elgin Twin Bars speak to this design relationship with outside manufacturers
Sears largest supplier from the early balloon years through about 1938 was Westfield Manufacturing, the company that had evolved out of Colonel Albert Pope’s Columbia brand, which is recognized as the first major American producer of Ordinary or High-Wheel bicycles.
The 1935-1937 Bluebird (and the later 1938 model) was manufactured exclusively by Westfield Manufacturing for Sears. The bike was also exclusively sold by Sears. The original design for the Bluebird was heavily patented and, interestingly, is credited in separarate patents to different inventors. The assignee on the patents is Sear Roebuck rather than Westfield so it is likely the design was originated either independently and sold to Sears or was drafted up by the in-house crew and then forwarded to Westfield for construction.
While I can see reasons that one might see “Murray” in the Bluebird, at the time of the bicycle’s introduction Murray-Ohio was producing tricycles and pedal cars and had not yet entered the field of bicycle production. When Murray did begin to produce bicycles (late 1936), they quickly became a powerhouse in that field and ultimately pushed Westfield aside to second-string status in the lucrative Sears supply chain. I recently read that some of Murray’s success was thought to come from their willingness to engineer their product to a cost for Sears, effectively shutting out the competition.
While Westfield and then Murray were the prime manufacturers of Elgin and later J. C. Higgins bicycles there were many smaller players that held smaller supply contracts for shorter periods of time but that is another topic.