If I have Feather, the Redeemed on the battlefield, and target her with the support part of Shoulder to Shoulder, what happens? Would Shoulder to Shoulder exile as it resolves and return to my hand, or would it resolve normally?
Shoulder to Shoulder is an instant or sorcery spell that targets; it doesn't matter that the word target only appears in the reminder text.
114.1e Some keyword abilities, such as equip and provoke, represent targeted activated or triggered abilities. In those cases, the phrase “target [something]” appears in the rule for that keyword ability rather than in the ability itself. (The keyword’s reminder text will often contain the word “target.”) See rule 702, “Keyword Abilities.”
So Feather's ability applies to Shoulder to Shoulder, and it's exiled as it resolves.
There is no "support part" of Shoulder to Shoulder; either the entire spell targets or it doesn't. Shoulder to Shoulder does target, so if you cast it and target a creature you control, then Feather, the Redeemed's ability will trigger and cause it to be exiled and put back into your hand as it resolves.
The fact that Shoulder to Shoulder also does something else in addition to the Support effect is irrelevant.