Is there a way to concentrate the vitamin content of a particular food?
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.