In English you could say either "paper" or "sheet of paper" here. In Polish could you use "papier" instead of "kartek" here. With a slightly different meaning of course.
When talking about sheets of paper we usualy use kartki, bacuase 'by default' you mean kartki papieru. In English sheets could also mean bed sheets (prześcieradła) - probably that is the reason why this sentence doesn't look like that: 'We do not have white sheets' -> ambiguous.
Also Polish 'papiery' is informal for documents. And if someone 'ma żółte papiery' (has yellow papers) it means that this person is crazy, but in the negative meaning.
Is this because of the colour of the form that you need for involontary commitment?
biały is singular masculine nominative białych is plural feminine(not masculine personal) Genitive
I do not know if you already know that
all adjectives have to match gender, case and number with a noun
there are two genders in plural masculine personal and not masculine personal that contains masculine animated not personal nouns, masculine not animated nouns, feminine nouns and neuter nouns
nie ma needs genitive
we don't have sheets of white paper - means the same as white sheets of paper in English
a card (as in card games, credit card, etc.) is "karta".
kartka is usually a sheet of paper, it may also be a postcard (kartka pocztowa / pocztówka), but in such a sentence, I wouldn't even think of a postcard.