One of the major political parties in Australia is the Australian Labor Party (ALP). Why do they use the American spelling of 'Labor', instead of the more expected Australian spelling of 'Labour'?

This issue stems from a time when there was no standardised spelling and "labor" and "labour" were used interchangeably.

Indeed, the political party flip-flopped between both spellings at some of its early conferences.

[The] Federal Conference used the spelling "Labor" in 1902, "Labour" in 1905 and 1908, and then "Labor" from 1912 onwards.

https://en.wikipedia.org/wiki/Australian_Labor_Party#Name_and_spelling

The party became the "Australian Labour Party" in 1908, then in 1918 named its branches as part of the "Australian Labor Party." It took some time (decades) before there was any uniformity of spelling.

"the way the spelling of 'Labor Party' was consolidated had more to do with the chap who ended up being in charge of printing the federal conference report than any other reason"

https://en.wikipedia.org/wiki/Australian_Labor_Party#Name_and_spelling