I don't think there's another NCAA sport where people other than players, coaches, and trainers are on the field of play.
But how would a caddie system even work?
Would the teams pay for caddies at $100-$150 per player, per round, per tournament? That would be $2,000 to $3,000 per team per regular tournament, or much more for a longer NCAA Championship. I know Oklahoma State has money, but Sam Houston State might not.
And most of the courses they play at don't have caddies anyway. So who would the caddies be? The other players on the team (so schools pay for for another five guys to fly, stay in the hotel, eat, etc., while missing school)? Or would the players have to find whoever is willing to pay their own way—parents, siblings, friends? That doesn't seem good for competitive equity.
There are some tournaments, like the occasional Stanford match play tournament at Cypress Point, where the players have to take caddies because of where they're playing. The teams pay for those. But again, those are small events at clubs that have a caddy program. I don't think Grayhawk or La Costa have 156 caddies ready to go.
In short, I don't see how it could possibly function in a good way.