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First of all, microwaves emit non-ionizing radiation and thus can't affect anything biological except for heating it. Wifi uses the same radiation and we bathe in that all the time, it is totally safe.
Second of all, even if the microwave blasted huge amounts of ionizing radiation into the underwear or even food when heating, it will NOT turn the food radioactive. This is a common misconception and radiation is often used to treat food in order to kill all the microbes, this makes the food very resistant to spoiling. Often MREs used in the millitary are treated this way to give them a large shelf life, which is a really good feature to have. The food is totally safe to eat (at least from a radiation standpoint).
It is possible to use radiation to make something radioactive, but the energy levels need to be extremely high. Think inside of nuclear reactors high. This is actually a large reason for radioactive waste, not just the spent fuel, but a lot of parts used inside the reactor become radioactive. This is why nuclear fusion reactors still output a lot of radioactive waste, as parts used inside the reactor (especially the shields) become radioactive.
However it's important to know radioactivity can absolutely spread and contaminate stuff. This is not because of the radiation, but because of the material itself spreading. It's usually in the form of dust that spreads, gets stuck all over the place and especially when it gets inside of biologics it can be a real problem. But the exact same risk exists for toxic materials that aren't radioactive, for example asbestos fibres.
Do the parts of a reactor become radioactive because of bombardment with ionizing electromagnetic waves, or because of bombardment with particles from the reaction that become embedded in them building up over time?
Several reasons. Outside of the reactor core itself, almost all items that “become” radioactive are just contaminated with radioactive waste. The item itself hasn’t turned radioactive, it’s just covered in radioactive particles that are usually very difficult to remove.
However, when you expose something to very high levels of neutron radiation (like you would find in a reactor core), it is possible to turn the atoms of the item itself radioactive. This is known as “activating”, and it’s how we produce many types of radiopharmaceutical drugs and other research isotopes. The amount of neutrons required to do this are basically nonexistent outside of nuclear reactors and particle accelerators, so it’s usually not a concern.
Source: my job is to put things into a nuclear reactor and turn them radioactive.
I think the confusion comes from using the same word for electromagnetic radiation and also for nuclear radiation which has actual alpha beta or gamma particles flying out.