Googled both trolley and buggy. I can't tell the difference. I call it a golf bag cart. I bought in the mid 1980s. The serial tag says it's a Powakaddy Classic.
Most Powakaddy gearboxes use a grease.
I would think any grease would do as it’s not really getting to high temperatures.
Just clean it first so there is no residue left.
Try YouTube there are lots of trolley maintenance vids on there.