Hello Tony
You are correct in thinking that enzyme concentration will directly influence Vmax; the more enzyme you have, the more substrate that can be processed essentially.
Unfortunately, there is no good intuition for Km. You may have to just remember that Km is an inherent property of the enzyme and is independent of enzyme or substrate concentration. Km is derived mathematically from the rate constants of the forward and reverse reactions forming the enzyme-substrate complex. I have included a link of the derivation (way beyond the scope of the MCAT) mainly so you can see it is quite a bit more complicated than your (logical) reasoning above.
Key takeaways: Km is an inherent property of the enzyme (independent of enzyme or substrate concentration). A low Km means the enzyme has a high affinity for substrate and vice versa. Vmax is the maximal rate of a reaction and occurs when the enzyme is saturated.
Hope that helps!
http://imgur.com/rupRg8j