I am playing an artificer. My mechanical servant is a large creature. My question is if I am riding my mount (which is 10ft. by 10ft.) and I attack a creature adjacent to my mount with a ranged attack, does it have disadvantage or not?
If you are within 5 feet it does, if you aren't it doesn't.
Being mounted does not change your size so your mount occupies a 10x10-foot space, you occupy a 5x5-foot space within your mount's space. If you are playing on a grid this has to be one of the 4 corner spaces of your mount's space - if so, the 5 of the 12 surrounding squares are within 5 feet and 7 aren't. If you don't play on a grid you could choose to be in the centre - if so all of the 12 surrounding squares are within 5 feet.
You can use your movement to change position on your mount but this may trigger Opportunity Attacks if you move out of an enemy's reach.