Logic would say that, if you're mounted on, say, an elephant, you wouldn't be able to swing a sword at a goblin at the elephant's feet.
But is there an official rule that allows/prevents non-reach melee weapon attacks while on a Huge (or bigger) mount?
However, you are first and foremost going to be limited by your reach. A huge creature is a 3x3 square assuming you are using a grid. If you are a medium creature and you are in the center square of your mount, then you would definitely need to use a reach weapon to hit anything adjacent to your mount.
It is worth noting that Jeremy Crawford has ruled and riding a mount does not change the effective size of your character:
Being mounted doesn't change your character's size.
Jeremy Crawford also said that when using a grid and miniatures you can "fit" the rider in any square occupied by the mount and that you can move around on your mount using your movement (here).
Regardless of the mechanism, if your character can be positioned on the mount such that it might be within reach of your weapon RAW you should be able to hit something with a non-reach melee attack.
However, the DM may adjudicate cases where this may or may not make sense to apply or what kind of checks or (dis)advantages may be involved to attempt such things.
For example you could try roping yourself to an elephant and leaning way off the side of it allowing you to hit things with a sword or some other non-reach weapon, but as a DM I certainly would not allow this to automatically succeed and would require some checks.
