This is a valid sentence grammatically. The sentence structure is: Subject + Adverb of manner + Verb Phrase.
Please note: 躺在草地上 is a whole piece in terms of sentence structure. Yes, it contains two parts: verb 躺 and location complement 在草地上, but these 2 parts are at the 2nd level, different from 他 or 静静地 as they are at the 1st level. That is why we can't say 他在草地上静静地躺, because it is not a fully finished sentence (except in poems though, you know poems have full of unfinished sentences).
I said 躺在草地上 is a whole piece, also because 躺 itself strongly demands a location complement. We can replace 躺 with other verbs which strongly demand location complement as well, like 走 在草地上 or 倒 在草地上. But if we replace it with verbs that don't strongly demand location complement, that makes it sounds awkward or foreign, like 听 在草地上, 看 在草地上, or 闻 在草地上.
As mentioned in section A, 他在草地上静静地躺 is not a finished sentence. It sounds very foreign. Instead we should say 他在草地上静静地躺 着. The reason I think is that Mandarin language employs a lot 2-syllable words to convey the sense of completeness. So using 躺 alone as verb in a sentence just gives people a strong sense of incompleteness. This is somewhat similar to the concept of Resolution in western tonal music theory, which means the move of a note or chord from dissonance (an unstable sound) to a consonance (a more final or stable sounding one). So we use 躺着 to make it complete.
This is a valid sentence grammatically. The sentence structure is: Subject + Adverb of place + Adverb of manner + Verb.
Note here I use the term adverb of place instead of location complement. The difference is that adverb of place is loosely related to the Verb, not a part of the Verb itself, while location complement is part of the Verb, whose function is to complete the Verb.
Be careful with this type of sentences: Subject + Adverb of place + Verb as they may not be able to converted to Subject + [Verb + location complement]. For example: 他在草地上看书 is good, but it really sounds awkward if you say 他看书在草地上.
Sentence A 他静静地躺在草地上 focuses on the action of 躺在草地上 as it is a whole piece, while sentence C 他在草地上静静地躺着 focuses on the action of 躺着 where the location of 草地 is not part of it.
P.S. Complements (Result / Potential / Direction / Degree / State / Quantity / Location / Time) are a very important feature of Chinese language grammar (See more details here Chinese Language Grammar - Complement), and whether you can use complements correctly determines your level of your Chinese language.
P.P.S English is a Subject + Verb language, while Chinese doesn't have clearly defined Verb in sentences. Actually Chinese is a Topic + Comment language, like 他的个子1米8. 他的个子 is topic, and 1米8 is comment. Another example: 他宽额头，高鼻梁，高颧骨，大眼睛。他 is topic, and 宽额头,高鼻梁,高颧骨,大眼睛 are all comments.