The short answer is: it depends on the bearing type inside, not the outer material.
Stainless steel is the housing and shank material. The sliding contact surface inside the rod end is a separate story. If that inner surface is metal-to-metal (steel ball against steel race), it needs lubrication. If it has a PTFE liner, it is self-lubricating and adding grease is unnecessary and sometimes counterproductive.
Where engineers get this wrong is assuming that because the part is stainless, it handles itself. It does not. A 316L stainless rod end with a steel/steel sliding surface will wear out quickly without relubrication, regardless of how corrosion-resistant the outer housing is. Corrosion resistance and lubrication are two independent performance dimensions.
There is also a thread galling issue specific to stainless. Austenitic stainless grades (304, 316) have a relatively high friction coefficient against themselves. During installation, if you thread a stainless rod end into a stainless rod without any lubricant or anti-seize, the threads can gall and lock permanently. This is a well-documented failure mode that has nothing to do with the bearing surface. Apply a thin coat of anti-seize compound to the thread before assembly. This is the step most maintenance guides skip.
The rule of thumb: Metal-to-metal contact surface requires periodic relubrication. PTFE-lined surface requires no grease but still needs visual inspection for liner wear. All stainless threaded installations require anti-seize at assembly.