Ordering Cake from a non-Jewish Bakery

May wedding cakes, birthday cakes, etc. be ordered from a non-Jewish bakery if the ingredients of the cake have been checked and are OK, or do we need to worry about their keilim? Could a dairy free cake from such a bakery be eaten after as dessert after a meat meal?
Background: a local bakery that specializes in cakes and cupcakes. They have a menu of cake and icing flavors from which customers choose (both dairy and non-dairy, no meat products AFAIK). These are then custom baked.

Not a problem