This quote is linked to the brother's plan for Doodle to become as physically robust as other boys before attending school for the first time. The brother hopes to avoid being embarrassed by the physically challenged Doodle and so embarks on a rigorous campaign to make Doodle stronger. Unfortunately, a series of illnesses plague Doodle and he is never able to meet the goals set by his brother. Rather than give up, however, the brothers continue:
The excitement of our program had now been gone for weeks, but still we kept on with a tired doggedness. It was too late to turn back, for we had both wandered too far into a net of expectations and had left no crumbs behind.
The brother is referring to the German fairy tale about Hansel and Gretel whose father tries to abandon them in the woods because he doesn't have enough food for them. The first time he attempts to lose them, Hansel leaves a trail of pebbles that lead back to the house. The second time the father takes them far into the woods Hansel leaves bread crumbs behind, which are eaten by birds and so the children are lost. In "The Scarlet Ibis" this allusion simply means that the brother and Doodle can never give up because it is too important to the brother's pride that Doodle not be seen as an embarrassment. They have no trail to follow back to the point where the brother could except that he has a disabled brother. Eventually the brother's pride leads to Doodle's death.
I assume you mean to say "literary allusion" rather than "literal allusion." If so, the allusion in the quote you give is to the fairy tale "Hansel and Gretel."
In this fairy tale, the two siblings, Hansel and Gretel, are going to be left in the woods to die by their parents. This is happening because their parents are too poor to feed them. As their father leads them into the forest, Hansel leaves a trail of bread crumbs so that he and his sister will be able to find their way home.
This trail of bread crumbs is what your quote is alluding to.