How could the USA have joined the Axis in WWII?
There would have to be a breakdown in constitutional government in the US in the 1930s - driven by an economic crisis so bad that a fascist demagogue comes to power.
That's what happened in Germany. However...
The demagogue in question was a
demagog. There was no one like that in the US as far as anyone knows.
The German political system was fluid and unstable. There had been a revolution and constitutional change only a few years earlier, parliamentary government with proportional election is much more open to new parties and factions, and aside from the Social Democrats and the Zentrum there were no large stable established parties in Germany.
And even with all those advantages, Hitler just barely succeeded. The Nazi vote peaked in mid 1932 - it dropped 12% in the November election. The NSDAP was out of money - literally begging on street corners. But the NSDAP and KDP between them had over half the seats, which forced the entry of the NSDAP into government. If in November the NSDAP and KPD had won a dozen fewer seats, an all-the-other-parties coalition government could have been formed, and the Nazis are frozen out.
But getting back to the U.S... Assume that some malignant amalgam of Huey Long, Boss Tweed, and Joe Kennedy gets in.
Overt alliance with Germany and Italy is still unlikely. However, it could happen if foreign alliances break right. The US was at odds with Japan, and Germany could choose to side with the white nation, while Britain, desperate to shore up its position, renewed its alliance with Japan.
The US was pro-China, and Germany supported China, sending a military advisor group. China OTL had good relations with the USSR, but that could fail - especially if the USSR takes over Xinjiang (the warlord there was pro-Soviet).
If the Depression goes that far in the US, it may splash back into Europe. Possibly France breaks down, and a Communist government gets in there. If there were actual Communist-sponsored insurrections in Latin America, that might drive the US to reach out to other anti-Communist powers.
What would have happened afterwards?
Impossible to say without knowing how it happens.