I don't see how there could have been very many Sadducees after 70 CE, or how influential they could have been to Karaism (whenever you think it may have started), given that their teachings before 70 CE are said to have been "received but by a few" and that "they are able to do almost nothing of themselves; for when they become magistrates ... they addict themselves to the notions of the Pharisees, because the multitude would not otherwise bear them" (Ant. 18.1.4) and "have not the populace obsequious to them" (Ant. 13.10.6).
And since this was the situation before
70 CE, how many Sadducees could there have been, and how much influence could they have had, after
And while some Karaites do not believe in the resurrection, most do, and this is in keeping with the DSS but not with the teachings of the Sadducees. And the Karaites and the DSS were champions of the poor, unlike the Sadducees, who are said to have been "able to persuade none but the rich" (Ant. 13.10.6).
As Malamat notes about the early Karaite Daniel al-Kumisi:
Daniel expressed bitterness at the material poverty of his fellow Karaites. They hear 'the shame with which they are reproached and the saying that whoever abandons the words of the Rabbanites and their festivals and ordinances will perish in poverty and distress.' Yet, from his point of view, penury 'is the sign of those who fear the Lord in Exile' ... The failure of Karaism in the tenth century was not due to R. Saadiah's keen polemics but to the [ascetic] lifestyle it offered.
https://books.google.com/books?id=2kSov ... ty&f=false
So it's hard for me to see the Karaites as emerging from or being directly related to the Sadducees, given that the latter did not believe in the resurrection, only appealed to the rich, and didn't have much influence on Jews even before 70 CE.
I suppose they were similar in that they were non-rabbinic and revered only the written Torah and in that sense Sadducees and Karaites have been around "since God gave his laws to the Jewish people," as the modern Karaite Nehemia Gordon puts it on his website, but for the above reasons Karaites otherwise don't appear to be related to the Sadducees to me.
Karaism has been around since God gave his laws to the Jewish people. At first those who followed YHWH's laws were merely called "Righteous" and it was only in the 9th century CE that they came to be called Karaites. The question of why God's followers are today called Karaites is really a question of the origin of the other sects. At first there was no reason to label the righteous as a separate sect because there was only the one sect which consisted of the whole Jewish people. Throughout history a variety of sects appeared and it was only to distinguish the righteous from these other groups which caused them in different periods to take on such names as Sadducees, Boethusians, Ananites, and Karaites.
Tell me all that you know and I'll show you snow and rain.