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I'm pretty sure it's natural to say "百{ひゃく}円です" as a casual abbreviation of "一百{いっぴゃく}円です" in a conversation. Same with "千{せん}円" vs. "一千{いっせん}円". This is similar in English conversation where it is ok to say "a/uh hundred yen" instead of "one-hundred yen".
But what about "万円です" and "億円です"? Do both of those also sound natural, or should you only say "一万{いちまん}円です" and "一億{いちおく}円です"?
You must say 「一万円{いちまんえん}です」 and 「一億円{いちおくえん}です」 using the number 「一/1」. It is just a custom we have and adhere to and those customs die hard in any culture.
Saying those two phrases without using 「一/1」 will make one sound very unnatural. If I heard 「まんえん」 by itself without any context, I would definitely think of the word 「蔓延{まんえん}」 ("spread", "prevail", etc.). I would not think of "10,000 yen" at all.
In business, however, we often say 「一千円{いっせんえん}」 to mean "1,000 yen", so that should probably be remembered. Note that 「一」 is pronounced 「いっ」, and not 「いち」 here.
No, 百円 is not the abbreviation of 一百円. The basic rules students learn at school are:
See Wikipedia for examples. (Particularly, note that 1000 is always せん, not いっせん, according to elementary textbooks)
If you want to practice, use this paper for fourth graders.
If you are an advanced learner, you may have actually heard いちじゅう (10), いっぴゃくまん (1,000,000) and so on. But they are basically jargon used by bankers, brokers, military members and so on. They even say ふたじゅうふた instead of にじゅうに to avoid any confusion. It's like saying "alpha" instead of saying "A".
As an exception, for some reason, いっせん is already popular in daily conversations among ordinary adults. You can forget the textbook rule here and say いっせん whenever you feel like saying it explicitly. But never use いっぴゃく and いちじゅう; they are usable only in special contexts, but simply wrong otherwise.
