Who handles ammo? Is it an S4 or an S3 function to request and draw Ammo?

It is handled by the S3. First you have to realize the ammo is forcasted in a system called the Total Ammunition Management Information System (TAMIS). The way we figure this out is by forecasting the training required to meet our unit training plan. Once the BN the requests the ammo from the S3 shop forecast is approved and the ammo is allotted. The ammo for each training event is specified in the Standards in Weapons Training (STRAC), DA PAM 350–38. Once all of this lines up its ready for the unit to pick up.

FM6.0 CH2
Training
2-49. G-3
 Conductingtrainingwithinthecommand.
 Preparingtrainingguidanceforthecommander’sapproval.
 Identifyingtrainingrequirements,basedontheunitmissionessentialtasklistandtrainingstatus.
 Determiningrequirementsforandallocationoftrainingresources.
 Organizing and conducting internal schools, and obtaining and allocating quotas for external
schools.
 Conductingtraininginspections,tests,andevaluations.
 Maintainingtheunitreadinessstatusofeachunitinthecommand.
 Compilingtrainingrecordsandreports.

Look in AR350-1, DA PAm 350-85 and AR5-31. Are you FORSCOM or RC? The FORSCOM reg spells it out

I see you are USAR. Things I try and remember that are important for Range activities regarding the USAR:
1) NCO's need to have current local range SOP training (mainly the range NCOIC and OIC)
2) Ammo needs to be orders ahead of time in accordance with the local range time line
3) Current trained CLS soldiers need to be on hand and there needs to be a designated emergency vehicle
4) There needs to be a training RSO at lease every other lane (ideally)
5) DO NOT FORGET HAZMAT PLACARDS!!!!!!! No one is picking up ANY ammo and driving it away without proper PLACARDS on the vehicle.

That's just to name a few. Then you need targets, magazines, staple guns, staples, etc....

S3 will plan all that and S4 will supply all that.