I'm trying to convert a chair from The Sims 4, but when the chair is put together with a dining table the shadow doesn't move with it. Does anyone know what is happening?

What did you clone? The shadow not moving with the chair is a known issue with barstools - I've never seen that happen with a dining chair.
OR, in case I'm not completely understanding your post, is that where the shadow is no matter where you place the chair (like not in front of a table)? If so, you'd need to move the shadow mesh to the right place in your 3D editor.

I cloned the base game Cowboy's Caboose Chair, the shadow shows up in the correct place when the chair isn't attached to a table, when it is the chair moves foward but the shadow stays static.

Just now got a chance to play around in SimPE. Since that chair doesn't have any joints I honestly don't know why the shadow isn't behaving - like I said, I've never used that chair as a clone. The shadow is it's own subset and I'm wondering if that has something to do with it - without the obvious joints in the GMDC... maybe something in the BHAVs, I dunno. HOWEVER - it'll be pretty easy to fix this. Just use a different chair as a clone and all should be well. The Contempto Penn Station Side Chair is a good choice.

Thank you! I just tested the Cowboy's Caboose Chair in game and it behaves the same way the clone did, I didn't know that because I've never used it before lol. I'll be sure to test the base game objects in game before cloning for now on, and I'll be using the chair you recommended