I have a character saying this in a novel set in the 1920s. I suspect it was in use at that time, but I've been fooled before by this kind of thing. (I had to delete a sentence in which a 1920s character says, "It's out of left field" because that phrase is traceable only back to 1940 or so.)
Have you checked Google Ngrams? That might help.
The Ngram indicates the phrase what's your angle first appeared in the mid-1920's, so you should be okay. However, when using this tool for such verification purposes, it's important to remember that Ngram results are based off of written documents (magazines, journals, and books); there's a chance the expression may have been used conversationally before it made its way into print. Some sayings need to reach a critical mass of usage before writers start using the phase in their work.
Searching the Chronicling America newspaper archives (1836-1922) gave one result. This question was asked in a letter to the New-York Tribune of December 20, 1922:
Oxford English Dictionary
Whilst not the exact phrase, the OED's sense 9b is:
With pre-1930 quotations: