Yes you can eat fully cooked chicken sausage cold if you want too although if you can heat the fully cooked sausage up first it will taste much better.
Since the fully cooked chicken sausage is cooked already it's ready to eat but just needs heated up but eating the cold fully cooked chicken sausage is not gonna harm you.
I've eaten foods like that cold when the power was out and I didn't have a way to heat it up but it would've been better heated up before eating.