Can a plane with 2 sets of shorter wings fly? I am proposing a plane that has 2 short wings of the same size on either side of the fuselage (two in the front and two in the back). Rather than using the horizontal stabilizer to destroy lift, the wings in the back help create lift and adjust the pitch of the plane.

One problem that I thought of is the wingtip vortex created by the front wing hitting the back wings. I'm guessing this can be resolved by placing the back wings higher than the front wings (from a front view of the fuselage).

Could this work?
What you are proposing is called a "tandem wing", and it's been done before. A brief search should produce more than enough research papers about the advantages and disadvantages of this particular design.

The near-complete absence of multiple-wing designs in modern aviation should be a hint that the design has issues. Unless you particularly need the advantages of the design more than the drawbacks, there's no particular reason to use it.