Is there a way to concentrate the vitamin content of a particular food? I'm looking for a way to extract vitamins out of common food sources of, well, vitamins.

Let's take, as an example, Vitamin A which is commonly found in carrots. A carrot (weighing around 60g) should contain around 500µg of retinoic acid (RA). RA is not soluble in water so in principle if I just dehydrate the carrot (which has around 50g of water) I would be left with around 10g of matter that still contains those 500µg of RA. I think that, with this simple procedure, I would be concentrating the amount of RA in my sample.

But maybe (surely) there are better ways to achieve this, to extract/purify a particular vitamin from food. Any ideas?

Also, some Vitamins are, instead, soluble in water, so we would need to apply another method for those.

And, finally, I would like to accomplish this using simple (household, if possible) techniques. It doesn't matter if I don't reach 100% purity, not even 1%, but if I can extract a 1g sample from my original carrot that has most of the RA still in it that would be huge.
You are correct that retinoids are poorly soluble in water. Vitamins A (retinol), D (calciferol), E (tocopherol) and K (phylloquinone) have high hydrophobicity and therefore should be extracted using organic non-polar solvents.

All these vitamins can be identified using UV/Visible spectroscopy.

Since your focus is on retinoids I shall focus on that (but the principles remain the same for other hydrophobic biomolecules).

Note: carrot has beta-carotene and it is a precursor for retinol and not a retinoid itself. 


Crush and homogenise the tissue
With some plant tissues you may need to grind the frozen tissue using liquid nitrogen repeatedly. Tissues containing polyphenols and lignins would need additional treatment but not necessary for carrots. 
Precipitate proteins (optional; not required in your case).
Extract using solvents such as hexane or chloroform. Isopropanol has also been used for beta-carotene extraction from carrots (Fikselová et al. 2008)[1]. Gundersen (2006)[2] summarizes the methods of extraction and analysis of retinoids from animal tissue.
To purify, you need to perform chromatography. You can differentiate the retinoids using UV/visible spectroscopy. Beta-carotene is easy to detect; it is bright orange in colour and has an absorption maximum at 450nm.




References:

[1] . Fikselová M., Šilhár S., Mareček J., Frančáková H., Czech J. Food Sci., 2008, 26 (4), 268-274.
[2].  Gundersen T. E., J. Neurobiol., 2006, 66 (7), 631–644.