The title of your post suggests that you'd like to know when it's acceptable to start a sentence with an "-ing" word more generally, and not just in cases like the one you cite.
"Sitting" and "thinking" might be contractions, as @Kate Bunting says. This is a song, so these phrases might be standing on their own and not be parts of a complete sentence.
But it's also possible that they are part of a longer sentence. For example: "Sitting here wide awake, Thinking about when I last saw you, I see that I can't live without you." I've just looked up the lyrics to "Standing in the Dark" on Google (just go to Google, there's no specific URL), and indeed, that's what we have here.
Here are the first three lines of the song: "Sitting here wide awake/Thinking about when I last saw you/I know you're not far away".
This is, I believe, a case of a present participial phrase acting as an adverb. (See this reference: https://medium.com/@engtuto1/can-gerunds-be-also-used-as-adjectives-89e5698411f3.) The two present participles, "sitting" and "thinking" directly affect the main verb, "know". The actions they describe cause the singer/narrator to "know" something.
You asked if "Sit" and "Think" would be correct. Generally, they probably would not, unless the intent was to give a command or a suggestion.
But the song's next line actually starts with a verb in the simple present tense! "Close my eyes and I still see you/Lying here next to me". Here, this is indeed a contraction of "I close my eyes". It works because the proximity of the word "I" to the phrase "close my eyes" makes it clear that the word "Close" is not a command.
Going back to the title of your post, there are certainly other cases where it's correct to start a sentence with an "-ing" word. Probably the most common is when the "ing" word is a gerund, a verb that becomes a noun when "-ing" is added.
An example would be, "Sitting is something I enjoy doing."
Does this answer your questions?