I think you may be confusing some terminology.The FCC requires that the VE's be at least General Class and the VEC must be an Extra Class.
First, here are the FCC rules for who can administer which amateur radio license exam:
§ 97.509 Administering VE requirements.
(a) Each examination for an amateur operator license must be administered by a team of at least 3 VEs at an examination session coordinated by a VEC. The number of examinees at the session may be limited.(b) Each administering VE must:(1) Be accredited by the coordinating VEC;(2) Be at least 18 years of age;(3) Be a person who holds an amateur operator license of the class specified below:(i) Amateur Extra, Advanced or General Class in order to administer a Technician Class operator license examination;(ii) Amateur Extra or Advanced Class in order to administer a General Class operator license examination;(iii) Amateur Extra Class in order to administer an Amateur Extra Class operator license examination.(4) Not be a person whose grant of an amateur station license or amateur operator license has ever been revoked or suspended.
Second, the Volunteer Examiner Coordinator (VEC) is the over-arching group that coordinates examination sessions. The FCC has many requirements for VECs, but I don't see any license class requirements mentioned in the rules. See §97.519, 97.521, and 97.525. I think when you said "VEC" above, you were referring to the VE Session Manager:
§ 97.513 VE session manager requirements.
(a) A VE session manager may be selected by the VE team for each examination session. The VE session manager must be accredited as a VE by the same VEC that coordinates the examination session. The VE session manager may serve concurrently as an administering VE.(b) The VE session manager may carry on liaison functions between the VE team and the coordinating VEC.(c) The VE session manager may organize activities at an examination session.