Service number is just another item to facilitate identification but is not an absolute requirement. You do want to include his date of birth (if you have it) and as much other info (unit, etc.) as you can.
By the way, the 1st, 3d and 4th Ranger Battalions were destroyed at Anzio on 31 January 1944 (the 4th was later rebuilt, the 1st and 3d were discontinued). If your info is accurate your man would have been a member of one of these.
If your IDPF request involves the above ranger and you're not sure of which battalion just indicate that he was a member of "a Ranger Battalion". This may further help them locate the correct file.