The theory of evolution via selection does not require any kind of divine intervention. It proposes that mutations randomly occur within the DNA, this causes changes in the phenotype (e.g. allowing digestion of a common food source), these changes can be good or bad (beneficial or deleterious). As a result, those with "good" mutations achieve more reproductions (the fitness of those mutant allele carriers is higher) and those mutations spread through the population. You can not direct the evolution of your DNA because mutation is random, even if we artificially induce mutations, we can not direct what the mutation does to the phenotype (though we can sometimes induce specific mutations known to have certain phenotypic effects). Selection favours some phenotypes over others but can only work with the mutations it is presented with - it can not direct the evolution of animals with wheels if mutations that allow wheels to form do not occur.
From this sequence you can see the evolution of a phenotype is best explained in a most parsimonious way without the inclusion of divine intervention. It is the principles of Occam's Razor. Calling divine intervention in to this theory would add further, and unwarranted, complication to the model and we would then have to find strong evidence for the existence of a divine being, something which is still yet to happen in the eyes of most evolutionary biologists. Strong evidence has been found which suits the theory without divine intervention (including unnecessary and untestable components to a theory goes against the basic principles of science).
Selection itself, though largely involved in evolution, is not necessary for traits to evolve. The neutral theory proposed by Motoo Kimura and developed in the last few decades suggests that traits can evolve with out selection, via genetic drift. This means traits develop in populations purely due to random sampling of neutral (or near neutral) random alleles. The relative importance of Selectionist vs Neutralist theories remains hotly debated within evolutionary biology.
Explaining your imaginary scenario with a current theory of evolution: A new digestive enzyme
The scene: There is a population of 100 deer. 50 males and 50 females. There is a potential for variance in mating success and that mating success is defined solely by the strength of the male.
Selection affects a trait: Males that digest their food better grow bigger and stronger. Thus selection favours males that get the most out of their food. If the ability to digest is completely genetically determined by a single locus (for simplification lets say it is) and there is only one allelic variant (one version of the gene) in our first generation then mating success will be equal among the males.
A mutant arises: In the next generation a single male carries a mutation in that locus. It is a mutated version of the gene which makes a key digestive enzyme work more efficiently.
Selection acts: This male is 10% larger and stronger than other males, and therefore sires more offspring in the next generation than any other male. These next generation offspring (technically half if it was a single mutation in a diploid organism - the other half have the ancestral haplotype) have the mutant allele. Then those males with the mutant allele also get more of the matings and so on and so on.
The allele is then spreading through the population. It will continue until all members of the population have the same allele (fixation) or will start over if a new allele arises which is even better. That's all evolution is, no divine beings necessary.
What if the allele was neutral? If the gene for a digestive enzyme existed but was not necessary in an organisms diet it would not know it could go out and use that food source. It may not even be able to. For example, a single fruitfly carries a mutation which has no cost and allows it to digest lactose. It is unlikely to be able to utilize that allele, because dairy products are not part of it's diet and not available in the wild. The allele would only drift in the population, eventually being lost (most likely as it starts at low frequency) or becoming fixed in the population. You can see a genetic drift simulation I wrote in R here. You can set f=1 and then play around with population size, number of generations, and replicates to get a feel for this process.
Note it is worth considering the effects of pleiotropy (if an allele has a good effect on one trait but a negative effect on another - including traits in different environments) and linkage (if the good allele is close to some deleterious mutations in the DNA it will struggle to spread until it is separated by recombination).