I pick businesses by who gives me the best value factoring in convenience and competence. I normally leave opinions out of the equation.
If this is really true, do you consider yourself a moral person?
Our choices as consumers (and as business people, and in every other facet of our lives) have moral implications. Ignoring those implications means behaving negligently toward morality, which is itself immoral, IMO.
Now... I'm not saying that anyone who supports Chick-fil-A is necessarily immoral. Maybe you can see a way to justify supporting them - I can't - but pretending that your decision, whatever it is, doesn't have moral consequences and isn't an expression of your personal values is sticking your head in the sand.
Money spent at Chick-fil-A *will* go toward hurting and oppressing gay people. If you think that doing this to gay people is wrong, and you still buy from Chick-fil-A, then you are supporting things you claim to oppose, and this is something you'll have to reconcile.
Giving money to someone, whether it's a lobbying organization, a church, or a business, is tacit endorsement of what they do. Ignoring this doesn't mean you're in the clear morally.