The sentence you wrote is mostly correct and easy to understand! Here's the sentence with a tiny change to make it flow better:
Original: Koalas have a strong tummy and can eat poisonous eucalyptus leaves without getting sick.
Improved: Koalas have a strong tummy and can eat poisonous eucalyptus leaves without getting sick.
The grammar is correct. "Have a strong tummy" means their stomachs are strong, and the sentence clearly says that koalas can eat eucalyptus leaves that are poisonous to others but don't get sick from them. Good job!
If you want, you can also say:
Koalas have strong tummies and can eat poisonous eucalyptus leaves without getting sick.
This just changes "tummy" to the plural "tummies," which is also correct.