How do I make this work?

I mean, you can homebrew it, if you want to. Look at old lore to see what rings and skills make sense for the clan and families, making sure that you give the same equivalent options as for existing clans.
Come up with (a) school(s), for them to pick.

Honestly though, ask yourself and them what the narrative of this should be. Why is there a Spider clan? Are they Tainted and evil, like last time? Why are they tolerated? What does it add to your game?
If you got good answers for that, which fit the game you want, then you need to go to the efforts mentioned before but probably have a better idea of what the clan and school should look like.

Personally, I would recommend to not do it, if at all avoidable, but that's just me.

I agree with the Myrion and think the best approach is to step back and ask what the player wants from their character. There are plenty of ways to create tainted samurai toeing the line without creating a new clan or school from scratch. Honestly, the arc of slowly realizing you work for soulless monsters from a socially acceptable lineage could be an interesting story, without all the macabre accoutrements.

Path of Waves has rules for creating new schools (if they are balanced or not, it's another matter), but I agree with everyone above, if you are playing in this new timeline where the Spider Clan doesn't exist (yet?), then why are you playing this edition in particular? Just go back to 4th edition. Now, if the player only wants to play a Tainted character, take a look at the Shadowlands book to see how to implement this.

Hey now, no need to be rude. 5e has quite a bit going for it anyway over 4e.

Making a school according to PoW should be fine, by the way. I'm just wary of introducing the Spider.

Yeah, if it were me, I would just say no to the 4e version. The Spider clan existed because of a very bizarre set of circumstances. I encourage my players to have a hand in world creation, but creating an organized, politically relevant clan in the middle of a blasted hellscape, and having it be accepted by the Empire at large, is a bit much.

Now, there are plenty of options for a character that checks off a lot of those Spider boxes. He could be a maho-tsukai or Bloodspeaker from any of the clans. He could be an untainted member of a ronin family at the Edge of the Empire (no relation), applying for recognition as a minor clan, but secretly made up of descendants of the old Snake Clan (or maybe they still exist?). Perhaps they claim a loose kinship to the Scorpion, and lay claim to some forgotten territory Northwest of the Wall. Completely untainted, of course!

You need to know what the long and short term goals are for the player, the character, and the clan. What story does the player want to tell? How likely is he to betray the other characters? How about the character? He could actually be a pretty decent guy, just trying to earn enough Glory and Status to get his family recognized, perhaps not realizing the extent of their blasphemy.

The would-be clan is another story. They're probably trying for that long-term goal of bringing Fu Leng back to life and putting him on the throne. Or at least putting themselves there and establishing a dark empire. They're going to have to do a lot of cleaning when the Imperial auditors come to see if they're worthy of Minor Clan status.

There are plenty of ways to make it work, but "the Spider Clan exists, and they're allowed in the Empire and they don't get slaughtered on sight" would be too much to ask in my games.