You may acquire gold and other items from merchants simply by groveling. Begging is mostly used for roleplay as the merchants tend to be as poor as you but you never know what one generous merchant may give! The item received is based on your begging skill level.
The begging skill only works on Asgard Merchants as they are the only ones sympathetic to a person down on their luck.
There are no merchants who train begging. To raise the skill, you must use the skill! Begging only goes to 60.
To use begging, just activate the skill from the skill book (or set an action macro) and target a merchant. If you are successful, that merchant will place something in your backpack.