Risking the ire of the heroic moderators here, but 'weaning' is the act of coaxing a baby to thrive not on his or her mother's milk, but rather on other less ideal sources of nourishment. With this remark, Alice Roosevelt is suggesting this poor fellow looked as though he was weaned straight from what could not be better for a baby, mother's milk, onto instead eating pickles. Pickles are sour, and for a baby this switch would likely be a shock, and a disappointment, even aside from the aspect regarding the diminished nutrients. In other words, Alice is suggesting he looks as even at the start of his life he met with a sour and disappointing shock. You might also extrapolate from her remark she is suggesting he looks like someone who is capable of being sustained by pickles, that he looks like someone who could survive being moved from mother's milk onto instead eating sour pickles. That he looks like someone who could somehow thrive on sourness.

The Shorter Oxford English Dictionary of Historical Principles offers this definition of wean: 'to accustom a (child or young animal) to the loss of its mother's milk: to cause to cease to be suckled...to detach or alienate from some accustomed object of pursuit or enjoyment; to reconcile by degrees to the privation of something...'