The short answer is yes.
There is no reason not to vaccinate children, and in fact, there are plenty of benefits to doing so. Apart from protecting their own health, vaccinating children also reduces household and community spread of flu.
Children can be vaccinated from the age of 6 months with the standard flu jab given by injection and once they reach the age of 2 years, they have the option of having the nasal flu spray. Unfortunately, the nasal flu spray is not suitable for children under the age of 2 years.
Children who have the nasal flu spray and have not been vaccinated previously generally need a second dose of vaccine 4 weeks after the first, for full effect.