Q:

Did starvation or malnutrition occur in indigenous American societies?

Colonizers of the Americas often paid natives for work in food, and justified the certainty in doing so as an improvement over the supposedly hazardous, unpredictable lifestyle those natives previously led. For example, Fermin Francisco de Lasuén wrote that mission neophytes "are sure of daily sustenance, when before they lived from hand to mouth."
Of course, the native societies were relatively stable. People could  could call on a range of resources and social networks as needed. Was there a real risk of deprivation or starvation in those native societies, before they were disrupted?

A:

Hunter-Gathers
Absolutely all hunter-gatherers live "hand-to-mouth", malnutrition is common, and starvation is not just a "risk" - it is a permanent threat.
This should be obvious because they cannot effectively store excess food and thus are subject to the standard predator–prey model:
plenty of food --> 
population expansion -->
depletion of food sources -->
population contraction -->
plenty of food

native societies were relatively very stable

Medieval inner city looks like a modern college campus "safe zone" compared to the level of violence in "native societies".

People could could call on a wide range of resources

This is an exaggeration: a tribe hunting deer will probably have little to fall back on, especially in winter.

social networks

A tribe is a single social unit. They hunt together, they eat together, they starve together. Remember, they cannot store food! Yes, some have a better teepee or moccasins, but not food.
A neighboring tribe might fare better (unlikely, but possibly), but the distance kills cooperation. They are not likely to be bosom buddies (they attack each other to kidnap women all the time), and transporting food is very hard.
Agricultural Societies
These fared better, but not by much.
They did not have high-yield crops and draught animals (and thus the wheel).
The first meant that they still lived hand-to-mouth (even though sightly better than hunter-gatherers because grain store better than meat) and the second meant that a local crop failure (due to, e.g., a drought) could not be mitigated by import.
See Guns, Germs, and Steel for details.
Food Storage
Effective food storage is a relatively recent invention. 
Pre-industrial societies did not produce much excess food for storage and could not store if very well.
E.g., GurvenKaplan2007 mention

the case of an Nunamiut Eskimo group that perished in its entirety, having been snowed in without sufficient food supplies to survive through the winter. 

PS. I am not saying that if a man fails to catch his daily quota of fish/fowl/venison, then his family will starve tomorrow. They can survive on the combination of yesterday's catch and the wife's gathering.
Death from starvation was probably not an annual event.
However, hunger was.
PPS. Further reading:

Hunger and Work in a Savage Tribe

