The Lucky Feather increases your odds to get lucky in ? stages, but only slightly. It is only useful in ? stages. ? stages (risk) has 2 choices, gain/lose health or have a good or bad card. This may make the cards good (more often) and gain health (more often). However that doesn't mean it always gives you good luck, as it can still lead to back luck which depends.
It is an Uncommon card and can be found in any stages where you can obtain a card or purchased in the shop. If you have a Dave's Doggo card that has to do with ?s, it'll cancel this card and +2 charisma, so Lucky Feather is deactivated. This is a major gamble.