You can say:
Kütüphane kartına sahip olmadan kütüphaneye giremem.
Here kartına is dative (“to the card”) and sahip olmak + -madan gives “without having.”
Change the ending to second person:
Kütüphane kartı olmadan kütüphaneye giremezsin.
Yes. Turkish has -sız / -siz / -suz / -süz meaning “without.” You could say:
Kütüphaneye kartsız giremem (“I can’t enter the library cardless”) or just Kartsız giremem if “library” is clear. But kütüphane kartı olmadan is more common.
Turkish allows some flexibility, but the usual order is Adverbial phrase + Location + Verb. You could move kütüphane kartı olmadan after the verb for emphasis:
Kütüphaneye giremem kütüphane kartı olmadan.
It’s grammatical but marked; the original order sounds most natural.