I have researched this question extensively myself, but not just on NutritionData or similar websites - in my local grocery store of all places. I keep finding surprisingly nutrient dense foods there, even though it is not a health food store. I believe you can get just about all of your vitamins and minerals from food, although some like magnesium will take more effort to find than others.
For example, I was shocked to discover that the lowly white potato (apparently not so lowly) contains 21% of the DV for magnesium and 46% of the DV for potassium. You don't even need to eat any fruit if you don't want to with those potassium levels (1 white potato has more potassium than a banana). Recently I found 100% cacao at my local grocery store, and it also is loaded with magnesium and several other minerals like iron, while having 0g sugar and plenty of other nutrients like fiber and saturated fat (24% of the DV for magnesium). It is intended for baking but I could care less as it is ridiculously cheap compared to products like cacao nibs that you find at health food stores and it is the exact same food.
Iodine you can get easily from most seafood, especially kelp, and those same foods usually have plenty of selenium as well. If you can't find kelp locally or don't like the taste, try kelp powder as a mineral supplement. My local grocery store has it or you can buy it online. Sea salt added to every meal will also give you more than adequate amounts of iodine and other trace minerals. I prefer the pink Himalayan variety for taste.
Another surprisingly nutrient dense food I found for a few dollars at my local grocery store is blackstrap molasses. It has 20% DV for calcium and iron, 10% DV for magnesium, and 17% of the DV for potassium per tbsp, with only 11 g sugar. Now that is a multimineral supplement, and it tastes amazing, so you can add it to almost any meal or baked product to make it taste better.
So yes you can get most if not all of your minerals (and vitamins) from foods if you look hard enough and do the research. For vitamins just check the WAPF website for the foods highest in them, but liver, eggs, grassfed butter, and cod liver oil contain the highest levels from what I have read.
References: 100% cacao: http://nutritiondata.self.com/facts/sweets/5390/2
White potatoes: http://nutritiondata.self.com/facts/vegetables-and-vegetable-products/2770/2