Why do non-accelerating objects exert force on each other? The equation for force is $\vec{F} = m\vec{a}$, where $\vec{a}$ is acceleration. Acceleration is a change in velocity. However, if an object with constant velocity (i.e. 0 acceleration) hits another object, it still exerts a force on it.
Why is this? Is there something fundamental that I'm missing here?
 A: $\vec{F} = m\vec{a}$ means that an object with a force $\vec{F}$ exerted upon it accelerates by an amount $\vec{a}$, not that an object accelerating with $\vec{a}$ exerts a force $\vec{F}$ on something else. Typically, the force exerted by an object has nothing to do with Newton's second law, but is given by other laws (like Coulomb's law in electrostatics).
But then, you have Newton's third law: every action has an equal and opposite reaction. This means that in a collision, the object that hits another also has an equal and opposite (instantaneous - or not, depending on the duration of collision) force exerted on it by the other. It does accelerate when it hits - it's velocity is only constant up to that point. And this acceleration is given by $\vec{a'} = -\frac{\vec{F'}}{m}$, where $\vec{F'}$ is the force exerted by the object on the other; and therefore the other exerts an equal and opposite force $-\vec{F'}$ on it, which we can use in Newton's second law.
A: Although an object that moves with constant velocity has no acceleration, it has kinetic energy and it has momentum.
Acceleration is not a conserved quantity.  It is not passed from one object to another.  Momentum and energy, however, are conserved quantities that pass from one object to another.
If a moving object hits a target, kinetic energy will be transferred to the target and may cause the target to gain its own acceleration.  This would be an elastic collision.  The results do not depend on the acceleration of the moving object, but only on its energy and its momentum at the instant of impact.
If the target absorbs all the kinetic energy of the collision and is deformed, turning kinetic into potential energy, this would be an inelastic collision, where kinetic energy is lost, and the target gains no acceleration.
There's a saying in finance that if you want to understand what's happening, "follow the money".  In physics, you could say "follow the energy" (NOT the acceleration).
The momentum of a moving object is the product of its velocity times its mass.  This, also, can be transferred from one object to another, unlike acceleration.
A: 2 points: first, $F=ma$ describes the acceleration of an object due to the sum of all forces acting on the object.  If these forces are in different directions, they may partly or fully cancel each other out.  In the case where the object is not accelerating (so it's moving with constant speed in a constant direction, or it's not moving at all), the sum of forces acting on it must be zero.  ($F$ and $a$ should really be vectors $\vec{F}$ and $\vec{a}$).
Second, objects need not be moving to exert forces on each other.  For example, a book on a table exerts a downward force on the table while the table exerts an upward force on the book.  But if there are forces, why isn't the book accelerating?  It's because each force on the book is balanced by an opposite force.  For example, the upward force of the table pushing on the book is balanced by the force of gravity pulling the book down.  However, these two forces don't have to be equal and opposite: they just happen to be so if the book isn't accelerating.  For example, if you cause the table to accelerate upward, then the upward force of the table on the book exceeds the downward gravitational force on the book, and so the book accelerates up with the table.
By the way, the fact that the forces on a non-accelerating object cancel each other out has nothing to do with Newton's 3rd law.  The 3rd law says every force is one of an "action-reaction pair" between two objects A and B.  The force of the object A acting on object B is always equal and opposite to the force of object B acting on object A.  Where it gets confusing is that often forces can be equal and opposite even though they are not "action-reaction pairs," for example gravity pulling the book down and the table pushing the book up.  These are not paired forces because one is between book and earth while the other is between book and table.  The "partner" to the force of gravity pulling the book down is actually an equal force of the book pulling the earth up!  Of course this force doesn't produce a measurable acceleration because the earth's mass is so huge.  But in fact, every force is one of an action-reaction pair.  
