As long as the installed stock will not fold it should be 922(v) compliant. I understand some have Bayo. Lugs. So, if it does, this will also need to be removed.
Now, here is where you run into problems. 922(r) states that you cannot build an unimportable rifle from more than 10 imported parts. So, you will need a certain number of US Made Parts.
Here is the ATF list of 20 parts. You need to have less than 10 foreign parts from this list:
(1) Receiver
(2) Barrel
(3) Barrel Extension
(4) Trunion
(5) Muzzle Brake/Flash Hider
(6) Bolt
(7) Bolt Carrier
(8) Charging Handle (Operating Rod)
(9) Gas Piston
(10) Trigger Housing
(11) Trigger
(12) Hammer
(13) Sear
(14) Disconnector
(15) Buttstock
(16) Pistol Grip
(17) Handgaurds
(18) Magazine Body
(19) Magazine Follower
(20) Magazine Floorplate
View Quote
The following are a listing of Uzi Parts or parts that may be on an Uzi. I am not familiar, but I know a number of these aren't on Uzis.
(01) Receiver *
(02) Barrel *
(03) Barrel Extension
(04) Trunion *
(05) Bolt *
(06) Bolt Carrier
(07) Charging Handle (Operating Rod) *
(08) Trigger Housing *
(09) Trigger *
(10) Hammer *
(11) Sear/Disconnector *
(12) Buttstock *
(13) Pistol Grip *
(14) Handgaurds *
(15) Magazine Body *
(16) Magazine Follower *
(17) Magazine Floorplate *
View Quote
The Uzi definitely has 15 of the 17 (marked with a star). As far as Barrel Extension and Bolt Carrier, I really don't know if it has these.
If it doesn't you will need 5 US Parts. If it has 1 then you will need 6 US Parts. If it has both, then you will need 7 US Parts.
US Magazine = 3 Parts
US Receiver = 1 Part
US Stock - 1 Part
So, assuming it doesn't have a Bolt Carrier or Barrel Extension, a US Receiver, Stock, and Magazine will be enough to make it legal.