The Mining Guild is the best place for Free-to-Play to obtain coal. Since there is a large amount of coal scattered around the place, you will be able to fill your inventory with ores in a decent amount of time, even when it's crowded. To be able to enter the guild, you need to have 60 Mining. You can also enter if you have 59 Mining, as you can drink Dwarven Stout (available at Falador Inn) to increase your level by one. It will go back down to your base level over time, which means you'll have to drink another glass of Dwarven Stout if you wish to enter again.
Location
There are three ways you can enter the Mining Guild. The most efficient way to get there would be to go from the South-Eastern Falador Bank and go down the ladder in the little complex (Green Path) as it is closest to the bank. The other way would be to go from the entrance at Ice Mountain or down the ladder of a little house in the east of Falador. All the paths are linked to the Dwarven Mine.
Rocks
There are 35 Coal and 5 Mithril rocks available. The ladders to the west brings you back near the South-Eastern Falador Bank. The door in the North-East leads to the rest of the Dwarven Mine. The ladder to the west right after you've gone through the door leads you back to the little house on the east of Falador.