first of all it's only a theory and I know kishi would probably never put this in the actual story
well sasuke's novel reveals karin is worried about sakura because she fells like sasuke is not coming back to the village and gaiden tell us sakura and karin were searching together for sasuke, here is where I think it all happens
what is sarada's origin ?
we all know orochimaru most likely have enough genetic material from sasuke to make a clone and we know karin may be very rabid at sometimes so after realizing sasuke would not accept sakura karin couldn't help hersell but fell like it was her obligation to help sakura for saving her life and then karin stealed all of sasuke's genetic material from orochimaru and probably a strand of sakura's hair and secretally started developing an embryo but at end even though she wanted to help sakura she still loved and just wanted to give something from herself to him so karin just did put some of her own gene along with sakura gene on it
would sakura accept this ?
nope but remember that chapter where karin had some kind of drug to make sasuke sleep ? what if karin still has this drug and just used it to make sakura unconscious and then injected the embryo into sakura
how this worked at end ?
I really don't know but karin is astute and learned about drugs from orochimaru so maybe she somehow made sasuke get drunk and ended up sleeping with sakura and done, within some months sarada would be born from sakura and nobody would learn the truth
the umbilical cord at karin's desk
after helping sakura give birth to sarada karin couldn't help but fell like sarada was also her daughter but she couldn't tell the truth to sasuke and sakura and then she just asked sakura if she could keep the umbilical and as a sign of gratitude for helping in the search for sasuke sakura accepted to give it to karin