Would a cancerous (lymphoma, breast cancer, etc.) axillary lymph node shrink?

Yes: A robust immune system / immune modulating medicine can do it.

Only if body immune: system were to respond to a cancer versus continue to let it grow (as is necessary to get to a size detectable by any current means beyond accidental discovery under a microscopic or by one of the current tests which can partially detect a few cancers before very advanced, e.g. PSA. In practice) because this is huge change in behavior, extremely rare. Thus if node shrank, likely not a cancer.

No: A lymph node replaced with malignant cells growing as a primary lymphod tumor or metastatic from another site which has spread to axilla will not shrink without treatment, either chemo or RT. The node will enlarge further or if unchanged will spread to adjacent nodes. It does not have the potential to metastasize further to non lymphatic tissue such as liver or lung.