The last version of iOS for the 4 was 7.1.2. 7.1.2 was jailbroken shortly after 8.x and Apple is still signing 7.1.2 (for the 4).
So, yes. You can jailbreak. You'll want the Taig jailbreak for 7.1.2. Additionally, it's also possible to downgrade your 4 to earlier versions of iOS if you so desire.