No, you do not need a complete profile before applying. However, look at things from the Employer perspective, the less they know about how great you are, your experience, your education, the less likely they are to move your application forward.
We highly recommend you complete your profile.
The good news is that your profile is live, this means that unlike say a paper CV which you can't update as soon as you update your profile the Employer will have the latest copy of it.
We know it takes some time to complete a profile, but once done you can use it to apply for any role within JobFairy.