That may be true for the first few weeks of induction as the body goes through carb withdrawal and adjusts to the difference in diet, but vitamin deficiency has never been a problem for me with all the veggies I eat, nor has constipation.
As others have pointed out, changing the way you eat drastically can result in constipation or, I might add, diarrhea until the body readjusts. Such symptoms are usually only temporary...a few weeks at most.
The muscle cramps or aching is also something that may occur during the first few weeks when weight loss is rapid and/or you are urinating a lot; calcium, magnesium and potassium are all water-soluble vitamins and you tend to lose more of them when you are urinating frequently than when you are not. Again, though, the body usually readjusts within a few weeks making supplementation unnecessary for most after that point.
Having said all that, taking a daily multi-purpose vitamin isn't a bad idea for anyone. I'd be willing to bet that most people, even those following the standard food pyramid, don't get the minimum RDA of a lot of things, especially if their diets are high in starches and low in brightly colored veggies and fruits. The fruits and veggies allowed even on induction are some of the most nutrient-dense veggies there are, while grains and starches are generally nutrient-poor, so most people who switch to low carb and trading nutrient-poor foods for nutrient-dense ones and calorie for calorie (or carb for carb) are getting better nutrition than they were before.