A Google ngrams search shows that the term was rare until the late 1930s and its use fell off very rapidly in the late 1940s. It had a small revival in the late ’70s and early ’80s, when Americans began to fear that Japanese businesses were out-competing them.
A search of the examples in the Google Books corpus shows that most examples before the 1920s are simple abbreviations, usually followed by a period, such as, “Jap. Govt.”
In the 1920s, one sees the term appear as slang. For example, H.G. Wells’ The Land that Time Forgot, published in 1924, makes a not-very-convincing effort to write in a Californian voice:
Californians, as a rule, are familiar with ju-jitsu, and I especially had made a study of it for several years, both at school and at the gym of the Los Angeles Athletic Club, while recently I had, in my employ, a Jap who was a wonder at the art.
Since this is throwing a slang term into a sentence of British English to make it sound more superficially American, I’m hesitant to call it a good example of authentic usage.
There is one other novel from this period in Google’s corpus that uses the term, also of a servant.
A 1928 report by the US Department of Agriculture mentions
The Japanese Dwarf variety usually is referred to by the growers by the colloquial name “Jap Dwarf” and was formerly known also as Sterling Dwarf.
A 1925 register of Jersey cattle lists several prize cows with “Jap” in their names. Some unmistakably use it as a noun to refer to a person, such as “The Jap’s Owl.”
ETA: Thanks to @GEdgar for finding another example: the Montgomery Ward Catalog from 1898 onward frequently used “Jap” as an adjective to refer to silk, satin, pearls and so on, such as in this page from the 1920 catalog. Notably, the catalog was not trying to disparage the products it sold!, but to exoticize them.
At this time, it would be a stretch to call the term respectful, but it wasn’t considered an insult until war propaganda began using it as one.