Was the machine sitting for a long period of time before you got it? The fuel pump diaphragm may be hardened. The pickup tube screen in the tank may also be plugged. The tank should be at least half full for easier starting.

I would surmise that you had some crud that was floating around in the gas tank. When the fuel completely drained, it clogged the outlet/pick up tube,

Post the engine Brand, model, type etc. so we can figure out what type carb you have.

From the picture in the link you posted it appears to be a Briggs 80202.

I had a VERY similar looking MTD. The unit was available in 5 or 8 HP Briggs OR Tecumseh engines. It came with 4 engine manuals!
IF it's a Briggs, I'd expect it to be a 13x202 model.

It's a craftsman 5 hp, I should have said that from the start. Turns out my dad was saving 10 year old gas from an old car he was fixing up and didn't label the can -.-
Now I know why the lawnmower isn't working either. Tomorrow morning I will drain the tanks and use carb cleaner, if that doesn't work, i'll just make my dad pay for the damage, lol. Oh, and you can be sure that I'll be making that old gas dissapear.

Ten year old gas? Man, that must stink. Does your dad still have the first dollar he ever earned? LOL

If it's a Craftsman engine, then it's a Tecumseh.
The chipper was made by MTD even if it has a Craftsman name on it. Sears doesn't make their own stuff!
Of course, you could actually post the numbers instead of making everyone guess.

New spark plug and the spray carb cleaner worked, then I re-adjusted the screw that controls the gas flow.
I did the same to the lawnmower, except a new spark plug, but it wont start. I'll get a new plug for that on Monday.
Thanks