I need to recreate this 15° brush-like vector texture in Adobe Illustrator. Any suggestions on how I do this? I was thinking there must be a brush that creates this but I really do not know. Any ideas?
Most of that was more than likely just drawn by hand and that is what I would do personally. If you do want to use something like a brush, you can get a similar effect (or at least a starting point for you to manually tweak later) by using Illustrator's Warp Tool[s].
You can adjust the Warp Tool[s] options (inc. brush shape and size*) by double clicking on the tool in the tool panel; a long thin brush at an angle should get you close to what you need:
Something that may help, instead of using the warp tool at an angle; rotate your artwork and hold SHIFT while using the Warp Tool to constrain in to a 90 degree angle then rotate your artwork back to its original position.
A quick 1 minute example using only the Eraser and Warp Tools:
You can probably get much better results with a bit more time and effort (and different tool settings). There are areas in your example where you would probably need to warp another shape on top that you can then subtract (via Pathfinder). You'll also need to manually clean up a lot of the edges etc. but you get the idea.
* Tip: you can manually adjust the brush size and shape on the fly by holding alt and dragging.