You can get Dragonshield by beating Desaix at Zofia Gate in Chapter 1. This is a way to get the weapon in the earliest stage. Note that it is very tought to beat him and you are not required to do so to complete the quest.
How to defeat Desaix
1. Get Lightning Sword
You can find Lightning Sword in an operations room in Deliverance Hideout. It is very useful, so don't forget to get.
2. Get Faye's class to be Cleric
Cleric Faye learns Psychic, a white spell to regain a distant ally's HP, at level 6. With Silque and Faye's white magic, you can safely beat Desaix.
3. Keep Slayde away
Desaix withdraws once you defeat Slayde, so keep him away using a character like Clair.
Jedah drops (Chapter 4)
Be defeating Jedah in Chapter 4, he drops Dragonshield. As same to Desaix in Chapter 1, you are not required to defeat Jedah at this point, and he withdraws after certain turns.
How to beat Jedah
If Ragnarok deals critical damage, you might be able to beat Jedah wiht just one attack.
Jeadh negates every first three attacks and your fourth attack can only inflict damage to him. Use Celica's Ragnarok as your fourth attack.
Dragon Zombie drops
Dragon Zombie sometimes drops Dragonshield, the rate is low though. If you want to get more than one Dragonshield, visit Dragon Shrine to farm Zombie Dragon after Chapter 3.
By completing "Shield with no Equal" subquest, you can get Dragonshield as a reward. Visit Rigel Village to get the sub-quest and you will be asked to bring 4 Dragon Scales.