Is vision of the entire board something that improves over time
Yes, but you have to consciously work on getting better at it. Being aware of it and posting about it is a good sign.
should I continue solving as many puzzles as possible to improve?
Yeah, that's one of the way you can work on improving it.
At the beginning of a puzzle, one thing I would do when I was new to chess, is for all of my rooks, queen, and bishops, I would trace their movement paths with my eyes across the whole board. Even if it was through other pieces.
For example here are some arrows showing the queen and one of the rooks.

That way I forced myself to be aware of how a far away piece might be close to influencing something I'm interested in attacking.
I know basic opening principles and basic strategy, but I always get tunnel vision and I easily miss ranged pieces attacking from the extremes of the board (Even when doing tactics puzzles). Is vision of the entire board something that improves over time or should I continue solving as many puzzles as possible to improve?