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.)
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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: