"Can I get four spoons?"
「スプーンを四つ」 is not a proper expression in Japanese. The most appropriate counter word for spoons is "本". It should be 「スプーンを4本ください」, but this answer was marked wrong. Reported on Oct. 27, 2017.
Most of the first few of these are tricky that way, one just has to learn them. 一つ - ひとつ, 二つ - ふたつ, 三つ - みっつ, 四つ - よっつ, 五つ - いつつ, 六つ - むっつ, 七つ - ななつ, 八つ - やっつ, 九つ - ここのつ
Is order important in this type of sentence? Could you say " [spoons] [please]" as well as "[spoons]  [please]"?
Possible, but in that case the particle goes between the noun and the counter: スプーンを四つください
On windows you press the key which is to the right of zero, for example here I typed "s u p u [that key] n [space]" which gets me スプーン.