What are the objects of perceptual awareness? The intuitive answer is probably "physical objects" - we think our perceptions are perceptions of the world (Direct Realism). The Shallow Foundationalist relies on direct realism when he claims that perception can directly justify our beliefs about the external world.
The Argument from Illusion is meant to cast doubt on this. There are many ways our perceptions can play tricks on us - for example, objects half-immersed in water look bent. The real object is not bent though, so whatever it is that we are perceiving, cannot be the actual object. Instead, we merely perceive a mental object, or sense datum. The Deep Foundationalist insists that only beliefs about sense data (not the external world) can serve as foundational beliefs (though we might be able to infer from these to beliefs about the world).
This highlights the gap between appearance and reality, between evidence and truth, between what we experience and what is actually there. Our senses clearly do not provide us with infallible knowledge of the world. But it gets worse. For if all we perceive is sense data, not the world itself, then it would seem that our perceptions don't tell us anything about the outside world at all. We cannot justify beliefs about the external world solely from sense data.
We seem to have the following options:
1) Phenomenalism (anti-realism) - deny that the physical world exists. Minds and sense-data are all there is. The realist structures described by science are just a "useful fiction" which help us to predict the order within our experiences.
2) Representational Realism - Suppose we add in the additional assumption that our sense data are usually caused by, and resemble, real objects. That would then allow us to make justified inferences about the external world (not infallible ones, but at least generally reliable).
[Either way, by basing all justification on our subjective sense impressions, the deep foundationalist seems committed to the existence of a Wittgensteinian private language, which could be a problem.]
#2 seems preferable, but how could we possibly know whether the assumption it relies upon is true or not? This is really just the problem of skepticism all over again. We would seem to have to resort to some form of externalism, whereby justification for our beliefs is independent of our awareness of the justification (e.g. possible worlds externalism, or perhaps reliabilism - i.e. the belief must be caused by a reliable process in fact, whether we realise it or not).
Alternatively, the internalist could accept that perception/experience does not justify our beliefs after all. Instead, it causes beliefs to form in the first place. Where does the justification come from, then? We would have to give up on foundationalism, and accept coherentism instead: no beliefs are intrinsically or categorically justified. Instead, a belief is justified to the extent that it coheres with our other beliefs.