Can you print vivid colors along with the color white using a heat press ?
You can "make t-shirts with vivid colors" using a heat press, but you can't print t-shirts with the press.
Like the sample that David posted above, plastisol transfers are a good way to get vibrant colors using a heat press.
Plastisol transfers aren't printed with your printer at home though; they are actually screen printing ink done by a screen printer, printed on transfer paper so they can be pressed on the shirt with a heat press.
Opaque transfer paper can be used to print on dark shirts, but the quality isn't as good as a plastisol printed transfer.
Here's a big long thread with more information on plastisol transfers: