Is it possible to make it through army basic training with posterior tibial tendonitis?

It depends... The severity of posterior tibial tendonitis will dictate your ability to complete the requirements with basic training. Mild cases can be treated with rehab, an orthotic and/or ankle brace along with anti-inflammatories. More advanced cases need more aggressive treatment and rest from high-impact activity often is requried. Your orthopaedic physician can help determine your ability to participate.
PT tendonitis. You could get through it but you will be in a lot of pain and can potentially tear the tendon during training.