Root Phone Without Unlocking Lockscreen
Short answer: Yes, it is possible. Different techniques can be used for this purpose (some of them are described e.g. in Andrew Hoog's book Android Forensics: Investigation, Analysis and Mobile Security for Google Android). It especially gets easy when USB Debugging is turned on.

Long answer: Please forgive me if I don't give this. Just a hint: There are exploits, and there are custom recovery images...