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Can the USSR remain at war without a Capital?

In my 1939 World at War Campaign, in sequence they lost Moscow, Kuybyshev and finally Perm (last alternative Capital), and now they don't have a capital on the map (and morale = 25), but they fight ...

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Yes, it can happen. But not for long. In most cases the USSR surrenders after the last capital has been taken. Sometimes surrender happens the turn after. Or, unlikely, the one after that.

I saw this questioned asked before, and if I remember the reply correctly in additional to all capitals and alternate capitals captured, the AI takes a look at unit losses and cities captured to determine surrender.

I'm playing a game where France got a second capital. Took Paris, but my human opponent has boatloads of French pieces. Capital moved to Buerdooux.

When you take over the last Capitol of a nation, they roll for surrender at each turn, they can remain in war 3% for each unit they have (6% for minors), so SU (or any nation) can fight forever in theory, in reality, this is a very low chance. SU usually has lots of units, if they have 33 or more, than they will remain in war, since the chance is over 100%, but as they lose more and more units, they will drop below, and will surrender rapidly.