Short answer: yes, you can use a plural form.
However, keep in mind the meaning of bottleneck: it's an issue limiting progress, while nothing else stands in the way of progress occurring. Imagine trying to build a house using a teaspoon to scoop up the cement: that would be a bottleneck! On the other hand, if you're out of bricks, cement and wood, then you just have problems, or in 'manager language', you have challenges to overcome.
The sentence does sound slightly odd to me: it sounds a bit like you need the bottlenecks to achieve your goals! More correct would be 'Here are the bottlenecks in our system to achieving this requirement' (using 'a bottleneck to' as a noun+preposition combination) or 'Here are bottlenecks in our system that prevent us from meeting this requirement'.
From what I see in online dictionaries, it is spelled bottlenecks or bottle-necks, not bottle necks.