The relevant sense comes under OED's definition section II, headed ...
To be active, to make efforts; to acquire or gather something energetically
...which is first recorded in 1835 as Get up, rouse and rustle about, and get away from these scores. Interestingly, that's the same year the sense Of livestock: to search for food, to forage; to graze was first recorded. The meaning To round up and steal (cattle, horses, etc.) wasn't recorded until 1886.
The exact sense to hunt out; (freq. in later use) to put together (a dish or meal). Now usu. with up. is first recorded (somewhat "marginally", I feel) in 1844 as He nailed my thumb in his jaws, and rostled [sic] up a handful of dirt & throwed it in my eyes. But OED's next citation isn't until 1890...
I was out one day after antelope (I ‘rustled’ all my meat, except a ham now and then as a luxury).
Given that the surreptitiously steal cattle sense was already "established" by then, I'm tempted to see that as closer to the intended sense above. But I think it's a good example of a usage showing part of the transition rustle = make a quiet noise => [quietly] gather together => gather together [food].
Note that OED only says this sense usually involves up. So, for example, I'll rustle some breakfast isn't actually "wrong" - it's just that most people would usually include the preposition.