Oof - the Norman Invasion was both massively influential and not at all. Scots was adapted from Anglian dialects, and the (very, very few) sources that are extant in Older Scots thus mimic Middle English of the same period. The first 'big' source for Scots is Barbour's Bruce, which was written in the 1370s and thus skews our interpretation of Scots linguistic history before this time. In essence, Scots relies on many of the same grammatical constructs as the post-Norman Middle English, so yes, the Norman Invasion had a massive influence on the development of Scots.

However, here is the other side of the coin: Scots developed independently from Middle English, and was definitely considered to be its own language in the fifteenth century, likely even earlier than that. While the two share a lot of vocabulary (some it derived from Old English), Scots takes its cues more from Gaelic, Norse, and Flemish, rather than Norman French. (It adopts vocabulary from France, but through contact with the mainland instead of through a French-speaking nobility)

If you're interested in medieval Scots, 'Older Scots: A Linguistic Reader' by Jeremy J. Smith is your first port of call, but if you have any other follow-up questions, I'd be happy to answer them!