Yes, it can. Back in the bad old days of early silicon diodes, avalanche wasn't typical, and diodes were rated by maximum voltage without breakdown (as opposed to minimum breakdown voltage at some bias current).
Processes weren't quite well enough controlled, and what happens is a microscopic defect breaks down, concentrating the entire applied current (~mA?) into a tiny area, which burns through and shorts the diode.
Anywhere you see a rating that applies current, you are guaranteed to have at least that much avalanche capability.
Although not necessarily more current than that. The defects might be less dramatic than the early diodes, but not so good as to avalanche the whole diode junction evenly. For that, there's actual avalanche rated parts. TVSs are made to break down over the whole die, and handle considerable energy.
Some rectifier diodes do carry an energy rating, 1-20mJ being fairly common. In the case of schottky diodes, this is absorbed by the guard ring, a PN junction that surrounds the schottky junction. The same may be true of other types, I don't know. This is why the energy rating might be quite small relative to the overall die size.
There are still plenty of diodes and transistors that can't handle avalanche. In the transistor case, you see V_CES ("collector-emitter sustain voltage"), rather than V_CEO (collector-emitter-open voltage, i.e., voltage applied C-E, with B open/unconnected).
I don't know of any IGBTs rated for avalanche. Probably for good reason, like parasitic SCR activation from putting free charges everywhere.
InGaN LEDs (i.e., blue, white, high efficiency green, etc.) don't seem to handle any avalanche, they die suddenly at 20-30V even with very low bias current. Older LED types (GaP, GaAs and alloys -- IR to green) usually handle a few mA, at random voltages (20 to >200V).
Tim