Do all Italian adjectives have to agree with the noun in gender and number? What if you include a color? For example, "a light green car" should be "una macchina verde chiara" but I cannot find any examples of the color verde used with the feminine form of the adjective chiaro.
Adjectives always have to agree.
The example you mention is actually very good, because it shows a mismatch in this sense.
You have to say
Una macchina verde chiaro
because chiaro here refers to "verde" and not to "macchina". So you need the masculine because of the agreement with verde itself, "verde chiaro" is a composite adjective.
In your example the color of the car is "light green", it's like you say "una macchina di color verde chiaro" or more shortly "una macchina verde chiaro".
I think it's simplier to understand with yellow: "una macchina gialla", but "una macchina (color) giallo chiaro".
But you can say "una tonalità (shade) verde chiara" even if "una tonalità (color) verde chiaro" is possible.