I wonder how you will explain yourself to God when the time comes...do you really think that supporting sin is acceptable to God? Do you believe that you are above God in your thinking and “morality”?
Their business isn’t a sin, and furthermore it is not my duty to punish sin in this world.
If you think that all the customers of this business are sinners by extension for buying their product and will have an ordeal explaining themselves to God, then frankly, I think something may be wrong with you.
These men have not sinned against me, they have made me delicious sandwiches is what they have done.
I will explain myself to God just fine, don’t you worry.
There's a difference between supporting sin and supporting a business run by sinners. If you refused to do the latter you'd have to go live off the grid in the woods and raise your own food, because everyone, whether they operate their own business or not, sins. Boycott the gay-owned sandwich shop, and your alternatives are the sandwich shop owned by the guy living with his girlfriend, or the shawarma stand owned by Muslims.
It's no wonder that when St. Paul reminded the Corinthian church not to associate with sexually immoral people, he immediately qualified that by saying he didn't mean people in the world, because you would have to leave the world entirely to avoid them (1 Cor. 5:9-10). In a pagan society, doing business with pagans is a practical necessity.
Chris said he didn't know what the politics of the shop owners were, so I infer that there is nothing in the shop openly promoting gay causes. So by buying lunch there, arguably you are not supporting homosexuality, you are purchasing goods and services. Which is different from a corporation that targets gays in their marketing: in that case, a portion of your money is going to promote immorality, and I'd say a boycott was arguably justified.
Matthew 5:16
Every single business I have ever dealt with has employed sinners.