Well, we could tackle the question from the other direction, of course. According to this Chinese (!) website, the Olympic podium was only introduced in 1932, copying the Empire Games at Hamilton, ONT two years before:
http://www.ebeijing....gin/t944420.htm
This picture, showing the first three in the 3 miles at Hamilton would seem to confirm that, with a rather 'agricultural-looking' construction, which might even be someone's school woodwork project.
The second-placed man is actually the Australian chap on the English winner's left, rather than what we'd expect today - and that salute to the crowd probably wouldn't have gone down so well a few years later:

This is the 1932 Olympics, with the silver medallists now on the winner's right.

Berlin 1936, silver medallists again on the winner's right. Front (rowing eights):

And back (men's long jump):

The podium seems to have been regarded as a dangerous Canadian innovation by the English and Australians, as I can't find any evidence that one was used in either the 1934 (London) or 1938 (Sydney) Empire Games! Maybe somebody got a splinter off the handrail?
Finally, in 1948, the British had the bright idea of putting numbers on it! Front and back.

The Swiss weren't that innovative for the winter games, though!

The only custom-built 'podium' I can recall in pre-war motor racing was at the Nürburgring, but - like the one at the rebuilt Monza, which wasn't used until the circuit re-opened after the war (at least for racing - General Mark Clark used it to review troops in 1945!) - it wasn't really a 'podium' as such. More like a raised verandah.