I'm not quite sure by your wording if the disposal is jammed or clogged. If the water runs out freely with no backup, we can eliminate the clogged drain issue. If the water does not run freely, a simple plunging (with a "sink only" plunger please!) may break the blockage free. If plunging still won't free up the flow, you're right to suspect you may have to drop the unit and snake the drain. When I do this stuff, I always use one of those disposable aluminum chaffing pans from the dollar store to catch the water and "mean/stinkin'/nasties" that are sure to fall out as you do your thing under there.
If your disposal isn't grinding/turning, then it's jammed. This is usually a simple task to free it up. When you got the disposal, it included an Allen (hex) wrench with the owners info. If you still have it, great! If not, no biggie...... A 3/8" Allen wrench will do.
On the BOTTOM of the disposal, dead center, you'll see where the Allen wrench fits into the end of the disposal's shaft. Rock that puppy back-and-forth until free movement is obtained. With your faucet running warm water at full throttle, fire up the disposal to send the offending materials to their grave.
35+ Years Pro Handyman and Home Improvement Contractor.