Yes, you can make your free attack from Improved Trip with a natural weapon you have already used that round.
There's no ironclad rule that you can only attack once in a round with a given natural weapon. The rule you're probably thinking of is that you don't get extra attacks with natural weapons from having high Base Attack Bonus, the way you get iterative attacks with manufactured weapons. From the SRD:
Creatures do not receive additional attacks from a high base attack bonus when using natural weapons.
This rule does not prevent creatures from attacking multiple times with a single natural weapon if they find some way of doing so other than iterative attacks from high BAB. Possibilities include:
- The Rapidstrike and Improved Rapidstrike feats from Draconomicon
- Having the ability to take multiple attack actions in a single round, like from the Shadow Pounce ability provided by Telflammar Shadowlord (Unapproachable East)
- Taking an attack of opportunity in a round when you have already attacked
- Getting extra attacks from the Cleave feat
The free attack from Improved Trip is an extra, triggered attack like those listed above - you're free to take it with any weapon, including a natural weapon that's already attacked.