Titan shifts can transform if their limbs are cut off, as with Erin's tranformation during the titan invasion in season 1. However, if a transformation has recently occurred, they are unable to transform until they have fully recovered.
The ability to transform into a Titan can be hindered by injuries inflicted on the human body. A human can transform into a Titan with injuries as severe as a missing limb, but only if a previous transformation has not occurred recently. If a human emerges from their Titan form and receives severe injury, this person cannot transform into a Titan again until said injuries have healed.
Power of the Titans
Erin also openly asked Reiner why he severed his limbs. Reiner responded to Erin's question by stating he did it because of the short time table he had after capturing him.
Seeing how they just had a battle not to long before this had happened, a transformation had just occurred. Severing Erin's limbs in this manner prevent him from transforming, because
- He recently transformed
- Received a serious injury shortly after coming out of his transformation