Are landlords allowed to charge a tenant extra if they want to sublet? My lease says an extra $100 will be charged per month for subletting.

If you signed and agreed to those terms it is. This is something you need to discuss before you lease. You can ask for it to be taken out or you won't sign the lease, then it is up to the landlord to rent or not to rent to you.

There are no state laws that prohibit a landlord from charging a fee if the tenant wants to sublet.

If that is what the terms of your contract state then you are LUCKY! In Texas we do not allow subletting under any circumstances. The lease would have to be terminated and termination fees would apply. Sometimes the termination fees total thousands of dollars and $100 is a deal for allowing you to sublet.