Can a soldier be chaptered for losing their clearance? If so, what kind of chapter would it be?

Yes. Absolutely.

But usually what we do is simply give you a mandatory reclass to an MOS that doesn't require a clearance.

You can be if your MOS requires a clearance and you can't be reclassed. If the event that triggered your loss of your clearance is derogative then yes. I have seen this due to a legal issues that were involved. If you were to lose your clearance due to debt then they could potential reclass you. If you lose your clearance due to committing a UCMJ chargeable offense then you might face separation.

Short answer yes. However you should get an option to reclass I. an MOS that does not require a clearance